Algebra - Quadratic Formula
Question 1
Find $a$, $b$ and $c$ given that the following are in the form $ax^2+bx+c=0$
a) $2x^2+3x+4=0$
a
$a=2$, $b=3$, $c=4$
Question ID: 20290010010
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b) $3x^2+5x+8=0$
a
$a=3$, $b=5$, $c=8$
Question ID: 20290010020
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c) $x^2+2x+5=0$
a
$a=1$, $b=2$, $c=5$
Question ID: 20290010030
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d) $4x^2-5x-8=0$
a
$a=4$, $b=-5$, $c=-8$
Question ID: 20290010040
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e) $-x^2-x+3=0$
a
$a=-1$, $b=-1$, $c=3$
Question ID: 20290010050
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f) $2x^2+6=0$
a
$a=2$, $b=0$, $c=6$
Question ID: 20290010060
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g) $x^2-3x=0$
a
$a=1$, $b=-3$, $c=0$
Question ID: 20290010070
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h) $x+2=0$
a
$a=0$, $b=1$, $c=2$
Question ID: 20290010080
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Question 2
Find $a$, $b$ and $c$ by rearranging into the form $ax^2+bx+c=0$
a) $4x^2+9x=2$
a
$a=4$, $b=9$, $c=-2$
Question ID: 20290020010
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b) $x^2=8x+3$
a
$a=1$, $b=-8$, $c=-3$
Question ID: 20290020020
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c) $x^2+6x=6x-2$
a
$a=1$, $b=0$, $c=2$
Question ID: 20290020030
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d) $24x^2-3=2(x^2+4)$
a
$a=22$, $b=0$, $c=-11$
Question ID: 20290020040
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e) $3x^2+1=\frac{5x+2}{3}$
a
$a=9$, $b=-5$, $c=1$
Question ID: 20290020050
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f) $4(9x-2)=\frac{3-x^2}{2x}$
a
$a=73$, $b=-16$, $c=-3$
Question ID: 20290020060
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g) $x^2(x-4)+3=x^3-\frac{6+2x}{5}$
a
$a=20$, $b=-2$, $c=-21$
Question ID: 20290020070
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h) $(2x+3)(9x-5)-3=\frac{x(6-3x)}{11}$
a
$a=201$, $b=181$, $c=-198$
Question ID: 20290020080
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Question 3
Insert the following into the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to find $x$
a) $a=1$, $b=10$, $c=9$
a
$x=-9$ or $x=-1$
Question ID: 20290030010
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b) $a=1$, $b=3$, $c=2$
a
$x=-1$ or $x=-2$
Question ID: 20290030020
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c) $a=1$, $b=5$, $c=6$
a
$x=-2$ or $x=-3$
Question ID: 20290030030
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d) $a=1$, $b=-5$, $c=6$
a
$x=2$ or $x=3$
Question ID: 20290030040
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e) $a=2$, $b=-14$, $c=20$
a
$x=2$ or $x=5$
Question ID: 20290030050
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f) $a=3$, $b=3$, $c=-18$
a
$x=3$ or $x=-2$
Question ID: 20290030060
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g) $a=1$, $b=12$, $c=36$
a
$x=-6$
Question ID: 20290030070
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h) $a=9$, $b=-54$, $c=81$
a
$x=3$
Question ID: 20290030080
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Question 4
Solve the following using the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
a) $x^2+7x+12=0$
a
$x=-3$ or $x=-4$
Question ID: 20290040010
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b) $x^2+10x+24=0$
a
$x=-4$ or $x=-6$
Question ID: 20290040020
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c) $x^2+21x=0$
a
$x=0$ or $x=-21$
Question ID: 20290040030
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d) $x^2-8x+15=0$
a
$x=3$ or $x=5$
Question ID: 20290040040
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e) $x^2-11x+28=0$
a
$x=4$ or $x=7$
Question ID: 20290040050
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f) $x^2-16=0$
a
$x=4$ or $x=-4$
Question ID: 20290040060
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g) $x^2-3x-10=0$
a
$x=5$ or $x=-2$
Question ID: 20290040070
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h) $x^2-14x-15=0$
a
$x=15$ or $x=-1$
Question ID: 20290040080
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Question 5
Solve the following using the quadratic formula
a) $2x^2-10x+12=0$
a
$x=2$ or $x=3$
Question ID: 20290050010
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b) $3x^2-3x-90=0$
a
$x=6$ or $x=-5$
Question ID: 20290050020
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c) $10x^2-60x+80=0$
a
$x=2$ or $x=4$
Question ID: 20290050030
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d) $8x^2-72=0$
a
$x=3$ or $x=-3$
Question ID: 20290050040
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e) $8x^2-8x-240=0$
a
$x=6$ or $x=-5$
Question ID: 20290050050
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f) $5x^2-25x=0$
a
$x=0$ or $x=5$
Question ID: 20290050060
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g) $42x^2-168x-1344=0$
a
$x=8$ or $x=-4$
Question ID: 20290050070
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h) $84x^2-168x-1260=0$
a
$x=5$ or $x=-3$
Question ID: 20290050080
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Question 6
Solve the following using the quadratic formula
a) $2x^2-5x-3=0$
a
$x=3$ or $x=-\frac12$
Question ID: 20290060010
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b) $3x^2-16x+5=0$
a
$x=5$ or $x=\frac13$
Question ID: 20290060020
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c) $5x^2+36x+7=0$
a
$x=-\frac15$ or $x=-7$
Question ID: 20290060030
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d) $8x^2-14x+3=0$
a
$x=\frac32$ or $x=\frac14$
Question ID: 20290060040
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e) $20x^2+13x-21=0$
a
$x=\frac34$ or $x=-\frac75$
Question ID: 20290060050
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f) $25x^2-9=0$
a
$x=\frac35$ or $x=-\frac35$
Question ID: 20290060060
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g) $6x^2+11x-7=0$
a
$x=\frac12$ or $x=-\frac73$
Question ID: 20290060070
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h) $45x^2-10x=0$
a
$x=0$ or $x=\frac29$
Question ID: 20290060080
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Question 7
Solve the following using the quadratic formula
a) $9x^2-12x+4=0$
a
$x=\frac23$
Question ID: 20290070010
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b) $25x^2+80x+64=0$
a
$x=-\frac85$
Question ID: 20290070020
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c) $4x^2-25=0$
a
$x=\pm\frac52$
Question ID: 20290070030
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d) $3x+7=0$
a
$x=-\frac73$
This is not a quadratic, so the quadratic formula fails
Question ID: 20290070040
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e) $15y^2-17y-110=0$
a
$y=\frac{10}{3}$ or $y=-\frac{11}{5}$
Question ID: 20290070050
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f) $25z^2-441=0$
a
$z=\pm\frac{21}{5}$
Question ID: 20290070060
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g) $8a-14=0$
a
$a=\frac74$
Question ID: 20290070070
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h) $16q^2-120q+225$
a
$q=\frac{15}{4}$
Question ID: 20290070080
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Question 8
Solve the following using the quadratic formula. Leave answers in exact form
a) $x^2+3x+1=0$
a
$x=\frac{-3+\sqrt{5}}{2},\:x=\frac{-3-\sqrt{5}}{2}$
Question ID: 20290080010
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b) $x^2+5x-1=0$
a
$x=\frac{-5+\sqrt{29}}{2},\:x=\frac{-5-\sqrt{29}}{2}$
Question ID: 20290080020
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c) $x^2+x-9=0$
a
$x=\frac{-1+\sqrt{37}}{2},\:x=\frac{-1-\sqrt{37}}{2}$
Question ID: 20290080030
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d) $x^2-x-4=0$
a
$x=\frac{1+\sqrt{17}}{2},\:x=\frac{1-\sqrt{17}}{2}$
Question ID: 20290080040
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e) $2x^2+3x-7=0$
a
$x=\frac{-3+\sqrt{65}}{4},\:x=\frac{-3-\sqrt{65}}{4}$
Question ID: 20290080050
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f) $3x^2-5x-3=0$
a
$x=\frac{5\pm\sqrt{61}}{6}$
Question ID: 20290080060
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g) $2x^2+x-11=0$
a
$x=\frac{-1\pm\sqrt{89}}{4}$
Question ID: 20290080070
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h) $-2x^2+x+8=0$
a
$x=-\frac{-1\pm\sqrt{65}}{4}$ or $x=\frac{1\pm\sqrt{65}}{4}$
Question ID: 20290080080
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Question 9
Solve the following using the quadratic formula. Leave answers as decimals to three decimal places
a) $x^2+7x-2=0$
a
$x=0.275$ or $x=-7.275$
Question ID: 20290090010
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b) $x^2-8x-11=0$
a
$x=9.196$ or $x=-1.196$
Question ID: 20290090020
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c) $x^2+x-13=0$
a
$x=3.140$ or $x=-4.140$
Question ID: 20290090030
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d) $4x^2+2x-5=0$
a
$x=0.896$ or $x=-1.396$
Question ID: 20290090040
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e) $2x^2-6x-1=0$
a
$x=3.158$ or $x=-0.158$
Question ID: 20290090050
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f) $3x^2-8x+2=0$
a
$x=2.387$ or $x=0.279$
Question ID: 20290090060
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g) $4x^2+10x+1=0$
a
$x=-0.104$ or $x=-2.396$
Question ID: 20290090070
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h) $6x^2-x-8=0$
a
$x=1.241$ or $x=-1.074$
Question ID: 20290090080
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Question 10
Solve the following using the quadratic formula. Leave answers in exact form, you need to simplify surds
a) $x^2+4x+2=0$
a
$x=\sqrt{2}-2,\:x=-2-\sqrt{2}$
Question ID: 20290100010
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b) $x^2+6x+3=0$
a
$x=\sqrt{6}-3,\:x=-3-\sqrt{6}$
Question ID: 20290100020
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c) $x^2-8x-2=0$
a
$x=4\pm3\sqrt{2}$
Question ID: 20290100030
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d) $2x^2-6x-5=0$
a
$x=\frac{3+\sqrt{19}}{2},\:x=\frac{3-\sqrt{19}}{2}$
Question ID: 20290100040
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e) $4x^2-8x+2=0$
a
$x=\frac{2\pm\sqrt{2}}{2}$
Question ID: 20290100050
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f) $7x^2+x-2=0$
a
$x=\frac{-1\pm\sqrt{57}}{14}$
Question ID: 20290100060
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g) $5x^2-10x+4=0$
a
$x=\frac{5+\sqrt{5}}{5},\:x=\frac{5-\sqrt{5}}{5}$
Question ID: 20290100070
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h) $-12x^2+12x+5=0$
a
$x=\frac{3\pm2\sqrt{6}}{6}$
Question ID: 20290100080
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Question 11
Solve the following using the quadratic formula. Leave answers in exact form, you may need to simplify surds
a) $2x^2-7x-13=-10$
a
$x=\frac{7\pm\sqrt{73}}{4}$
Question ID: 20290110010
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b) $2x^2-36=x$
a
$x=\frac{9}{2},\:x=-4$
Question ID: 20290110020
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c) $x^2-6x+7=32-3x^2-6x$
a
$x=\pm\frac{5}{2}$
Question ID: 20290110030
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d) $6y^2=94$
a
$y=\pm\sqrt{\frac{47}{3}}$
Question ID: 20290110040
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e) $\frac{1}{3}x^2+x-\frac{1}{3}=0$
a
$x=\frac{-3+\sqrt{13}}{2},\:x=\frac{-3-\sqrt{13}}{2}$
Question ID: 20290110050
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f) $0.4x^2+0.6x-0.5=0$
a
$x=\frac{-3\pm\sqrt{29}}{4}$
Question ID: 20290110060
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g) $5x^2+\frac{1}{2}x-0.2=0$
a
$x=\frac{-1\pm\sqrt{17}}{20}$
Question ID: 20290110070
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h) $\frac{1}{3}x^2+x-5=0.3x-2x^2-\frac{1}{8}$
a
$x=\frac{3\left(-7\pm3\sqrt{511}\right)}{140}$
Question ID: 20290110080
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Question 12
Answer the following
a) The solutions to a quadratic equation using the quadratic formula are $x=\frac{9+\sqrt{61}}{2}$ or $x=\frac{9-\sqrt{61}}{2}$. What was the quadratic equation?
a
$x^2-9x+5=0$
Question ID: 20290120010
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b) The solutions to a quadratic equation using the quadratic formula are $x=\frac{\sqrt{3}-3}{3},\:x=-\frac{3+\sqrt{3}}{3}$. What was the quadratic equation?
a
$3x^2+6x+2=0$
Question ID: 20290120020
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c) A rectangular room is $3$m longer than it is wide. The area of the lawn is $10$m$^2$. What are the dimensions of the room?
a
$2$m by $5$m
Question ID: 20290120030
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d) The product of two consecutive numbers is $4556$. What are the two numbers?
a
$67$ and $68$
Question ID: 20290120040
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e) Sam thinks of a number. He doubles it and adds 4 to it. He then multiplies this by his original number and gets $96$. What was his original number?
a
His original number was $6$
Question ID: 20290120050
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Question 13
Follow these steps to prove the quadratic formula
a) The general equation of a quadratic is $ax^2+bx+c=0$. Divide everything by $a$
a
$x^2+\frac bax+\frac ca=0$
Question ID: 20290130010
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b) What do you need to add to $x^2+\frac ba x$ to make the expression a perfect square?
a
Add $\frac{b^2}{4a^2}$ to make $x^2+\frac bax+\frac{b^2}{4a^2}$
Question ID: 20290130020
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c) Rewrite $x^2+\frac bax+\frac ca=0$ with the addition and subtraction of your answer to part b
a
$x^2+\frac bax+\frac{b^2}{4a^2}-\frac{b^2}{4a^2}+\frac ca=0$
Question ID: 20290130030
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d) Complete the square on your equation in part c
a
$\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a^2}+\frac ca=0$
Question ID: 20290130040
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e) Collect everything not in brackets to the right hand side of the equation
a
$\left(x+\frac{b}{2a}\right)^2=\frac{b^2}{4a^2}-\frac ca$
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f) Put everything on the right hand side of the equation into one fraction
a
$\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$
Question ID: 20290130060
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g) Square root everything. Remember there will be a positive and a negative!
a
$x+\frac{b}{2a}={\frac{\pm\sqrt{b^2-4ac}}{2a}}$
Question ID: 20290130070
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h) Make $x$ the subject. Everything on the right hand side should be in one fraction.
a
$x={\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$
Question ID: 20290130080
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Question 1
Find $a$, $b$ and $c$ given that the following are in the form $ax^2+bx+c=0$
a) $2x^2+3x+4=0$
b) $3x^2+5x+8=0$
c) $x^2+2x+5=0$
d) $4x^2-5x-8=0$
e) $-x^2-x+3=0$
f) $2x^2+6=0$
g) $x^2-3x=0$
h) $x+2=0$
Question 2
Find $a$, $b$ and $c$ by rearranging into the form $ax^2+bx+c=0$
a) $4x^2+9x=2$
b) $x^2=8x+3$
c) $x^2+6x=6x-2$
d) $24x^2-3=2(x^2+4)$
e) $3x^2+1=\frac{5x+2}{3}$
f) $4(9x-2)=\frac{3-x^2}{2x}$
g) $x^2(x-4)+3=x^3-\frac{6+2x}{5}$
h) $(2x+3)(9x-5)-3=\frac{x(6-3x)}{11}$
Question 3
Insert the following into the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to find $x$
a) $a=1$, $b=10$, $c=9$
b) $a=1$, $b=3$, $c=2$
c) $a=1$, $b=5$, $c=6$
d) $a=1$, $b=-5$, $c=6$
e) $a=2$, $b=-14$, $c=20$
f) $a=3$, $b=3$, $c=-18$
g) $a=1$, $b=12$, $c=36$
h) $a=9$, $b=-54$, $c=81$
Question 4
Solve the following using the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
a) $x^2+7x+12=0$
b) $x^2+10x+24=0$
c) $x^2+21x=0$
d) $x^2-8x+15=0$
e) $x^2-11x+28=0$
f) $x^2-16=0$
g) $x^2-3x-10=0$
h) $x^2-14x-15=0$
Question 5
Solve the following using the quadratic formula
a) $2x^2-10x+12=0$
b) $3x^2-3x-90=0$
c) $10x^2-60x+80=0$
d) $8x^2-72=0$
e) $8x^2-8x-240=0$
f) $5x^2-25x=0$
g) $42x^2-168x-1344=0$
h) $84x^2-168x-1260=0$
Question 6
Solve the following using the quadratic formula
a) $2x^2-5x-3=0$
b) $3x^2-16x+5=0$
c) $5x^2+36x+7=0$
d) $8x^2-14x+3=0$
e) $20x^2+13x-21=0$
f) $25x^2-9=0$
g) $6x^2+11x-7=0$
h) $45x^2-10x=0$
Question 7
Solve the following using the quadratic formula
a) $9x^2-12x+4=0$
b) $25x^2+80x+64=0$
c) $4x^2-25=0$
d) $3x+7=0$
e) $15y^2-17y-110=0$
f) $25z^2-441=0$
g) $8a-14=0$
h) $16q^2-120q+225$
Question 8
Solve the following using the quadratic formula. Leave answers in exact form
a) $x^2+3x+1=0$
b) $x^2+5x-1=0$
c) $x^2+x-9=0$
d) $x^2-x-4=0$
e) $2x^2+3x-7=0$
f) $3x^2-5x-3=0$
g) $2x^2+x-11=0$
h) $-2x^2+x+8=0$
Question 9
Solve the following using the quadratic formula. Leave answers as decimals to three decimal places
a) $x^2+7x-2=0$
b) $x^2-8x-11=0$
c) $x^2+x-13=0$
d) $4x^2+2x-5=0$
e) $2x^2-6x-1=0$
f) $3x^2-8x+2=0$
g) $4x^2+10x+1=0$
h) $6x^2-x-8=0$
Question 10
Solve the following using the quadratic formula. Leave answers in exact form, you need to simplify surds
a) $x^2+4x+2=0$
b) $x^2+6x+3=0$
c) $x^2-8x-2=0$
d) $2x^2-6x-5=0$
e) $4x^2-8x+2=0$
f) $7x^2+x-2=0$
g) $5x^2-10x+4=0$
h) $-12x^2+12x+5=0$
Question 11
Solve the following using the quadratic formula. Leave answers in exact form, you may need to simplify surds
a) $2x^2-7x-13=-10$
b) $2x^2-36=x$
c) $x^2-6x+7=32-3x^2-6x$
d) $6y^2=94$
e) $\frac{1}{3}x^2+x-\frac{1}{3}=0$
f) $0.4x^2+0.6x-0.5=0$
g) $5x^2+\frac{1}{2}x-0.2=0$
h) $\frac{1}{3}x^2+x-5=0.3x-2x^2-\frac{1}{8}$
Question 12
Answer the following
a) The solutions to a quadratic equation using the quadratic formula are $x=\frac{9+\sqrt{61}}{2}$ or $x=\frac{9-\sqrt{61}}{2}$. What was the quadratic equation?
b) The solutions to a quadratic equation using the quadratic formula are $x=\frac{\sqrt{3}-3}{3},\:x=-\frac{3+\sqrt{3}}{3}$. What was the quadratic equation?
c) A rectangular room is $3$m longer than it is wide. The area of the lawn is $10$m$^2$. What are the dimensions of the room?
d) The product of two consecutive numbers is $4556$. What are the two numbers?
e) Sam thinks of a number. He doubles it and adds 4 to it. He then multiplies this by his original number and gets $96$. What was his original number?
Question 13
Follow these steps to prove the quadratic formula
a) The general equation of a quadratic is $ax^2+bx+c=0$. Divide everything by $a$
b) What do you need to add to $x^2+\frac ba x$ to make the expression a perfect square?
c) Rewrite $x^2+\frac bax+\frac ca=0$ with the addition and subtraction of your answer to part b
d) Complete the square on your equation in part c
e) Collect everything not in brackets to the right hand side of the equation
f) Put everything on the right hand side of the equation into one fraction
g) Square root everything. Remember there will be a positive and a negative!
h) Make $x$ the subject. Everything on the right hand side should be in one fraction.
Answers
Question 1
a) $a=2$, $b=3$, $c=4$
b) $a=3$, $b=5$, $c=8$
c) $a=1$, $b=2$, $c=5$
d) $a=4$, $b=-5$, $c=-8$
e) $a=-1$, $b=-1$, $c=3$
f) $a=2$, $b=0$, $c=6$
g) $a=1$, $b=-3$, $c=0$
h) $a=0$, $b=1$, $c=2$
Question 2
a) $a=4$, $b=9$, $c=-2$
b) $a=1$, $b=-8$, $c=-3$
c) $a=1$, $b=0$, $c=2$
d) $a=22$, $b=0$, $c=-11$
e) $a=9$, $b=-5$, $c=1$
f) $a=73$, $b=-16$, $c=-3$
g) $a=20$, $b=-2$, $c=-21$
h) $a=201$, $b=181$, $c=-198$
Question 3
a) $x=-9$ or $x=-1$
b) $x=-1$ or $x=-2$
c) $x=-2$ or $x=-3$
d) $x=2$ or $x=3$
e) $x=2$ or $x=5$
f) $x=3$ or $x=-2$
g) $x=-6$
h) $x=3$
Question 4
a) $x=-3$ or $x=-4$
b) $x=-4$ or $x=-6$
c) $x=0$ or $x=-21$
d) $x=3$ or $x=5$
e) $x=4$ or $x=7$
f) $x=4$ or $x=-4$
g) $x=5$ or $x=-2$
h) $x=15$ or $x=-1$
Question 5
a) $x=2$ or $x=3$
b) $x=6$ or $x=-5$
c) $x=2$ or $x=4$
d) $x=3$ or $x=-3$
e) $x=6$ or $x=-5$
f) $x=0$ or $x=5$
g) $x=8$ or $x=-4$
h) $x=5$ or $x=-3$
Question 6
a) $x=3$ or $x=-\frac12$
b) $x=5$ or $x=\frac13$
c) $x=-\frac15$ or $x=-7$
d) $x=\frac32$ or $x=\frac14$
e) $x=\frac34$ or $x=-\frac75$
f) $x=\frac35$ or $x=-\frac35$
g) $x=\frac12$ or $x=-\frac73$
h) $x=0$ or $x=\frac29$
Question 7
a) $x=\frac23$
b) $x=-\frac85$
c) $x=\pm\frac52$
d) $x=-\frac73$
This is not a quadratic, so the quadratic formula fails
e) $y=\frac{10}{3}$ or $y=-\frac{11}{5}$
f) $z=\pm\frac{21}{5}$
g) $a=\frac74$
h) $q=\frac{15}{4}$
Question 8
a) $x=\frac{-3+\sqrt{5}}{2},\:x=\frac{-3-\sqrt{5}}{2}$
b) $x=\frac{-5+\sqrt{29}}{2},\:x=\frac{-5-\sqrt{29}}{2}$
c) $x=\frac{-1+\sqrt{37}}{2},\:x=\frac{-1-\sqrt{37}}{2}$
d) $x=\frac{1+\sqrt{17}}{2},\:x=\frac{1-\sqrt{17}}{2}$
e) $x=\frac{-3+\sqrt{65}}{4},\:x=\frac{-3-\sqrt{65}}{4}$
f) $x=\frac{5\pm\sqrt{61}}{6}$
g) $x=\frac{-1\pm\sqrt{89}}{4}$
h) $x=-\frac{-1\pm\sqrt{65}}{4}$ or $x=\frac{1\pm\sqrt{65}}{4}$
Question 9
a) $x=0.275$ or $x=-7.275$
b) $x=9.196$ or $x=-1.196$
c) $x=3.140$ or $x=-4.140$
d) $x=0.896$ or $x=-1.396$
e) $x=3.158$ or $x=-0.158$
f) $x=2.387$ or $x=0.279$
g) $x=-0.104$ or $x=-2.396$
h) $x=1.241$ or $x=-1.074$
Question 10
a) $x=\sqrt{2}-2,\:x=-2-\sqrt{2}$
b) $x=\sqrt{6}-3,\:x=-3-\sqrt{6}$
c) $x=4\pm3\sqrt{2}$
d) $x=\frac{3+\sqrt{19}}{2},\:x=\frac{3-\sqrt{19}}{2}$
e) $x=\frac{2\pm\sqrt{2}}{2}$
f) $x=\frac{-1\pm\sqrt{57}}{14}$
g) $x=\frac{5+\sqrt{5}}{5},\:x=\frac{5-\sqrt{5}}{5}$
h) $x=\frac{3\pm2\sqrt{6}}{6}$
Question 11
a) $x=\frac{7\pm\sqrt{73}}{4}$
b) $x=\frac{9}{2},\:x=-4$
c) $x=\pm\frac{5}{2}$
d) $y=\pm\sqrt{\frac{47}{3}}$
e) $x=\frac{-3+\sqrt{13}}{2},\:x=\frac{-3-\sqrt{13}}{2}$
f) $x=\frac{-3\pm\sqrt{29}}{4}$
g) $x=\frac{-1\pm\sqrt{17}}{20}$
h) $x=\frac{3\left(-7\pm3\sqrt{511}\right)}{140}$
Question 12
a) $x^2-9x+5=0$
b) $3x^2+6x+2=0$
c) $2$m by $5$m
d) $67$ and $68$
e) His original number was $6$
Question 13
a) $x^2+\frac bax+\frac ca=0$
b) Add $\frac{b^2}{4a^2}$ to make $x^2+\frac bax+\frac{b^2}{4a^2}$
c) $x^2+\frac bax+\frac{b^2}{4a^2}-\frac{b^2}{4a^2}+\frac ca=0$
d) $\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a^2}+\frac ca=0$
e) $\left(x+\frac{b}{2a}\right)^2=\frac{b^2}{4a^2}-\frac ca$
f) $\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$
g) $x+\frac{b}{2a}={\frac{\pm\sqrt{b^2-4ac}}{2a}}$
h) $x={\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$
Question 1
Find $a$, $b$ and $c$ given that the following are in the form $ax^2+bx+c=0$
a) $2x^2+3x+4=0$
b) $3x^2+5x+8=0$
c) $x^2+2x+5=0$
d) $4x^2-5x-8=0$
e) $-x^2-x+3=0$
f) $2x^2+6=0$
g) $x^2-3x=0$
h) $x+2=0$
Question 2
Find $a$, $b$ and $c$ by rearranging into the form $ax^2+bx+c=0$
a) $4x^2+9x=2$
b) $x^2=8x+3$
c) $x^2+6x=6x-2$
d) $24x^2-3=2(x^2+4)$
e) $3x^2+1=\frac{5x+2}{3}$
f) $4(9x-2)=\frac{3-x^2}{2x}$
g) $x^2(x-4)+3=x^3-\frac{6+2x}{5}$
h) $(2x+3)(9x-5)-3=\frac{x(6-3x)}{11}$
Question 3
Insert the following into the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to find $x$
a) $a=1$, $b=10$, $c=9$
b) $a=1$, $b=3$, $c=2$
c) $a=1$, $b=5$, $c=6$
d) $a=1$, $b=-5$, $c=6$
e) $a=2$, $b=-14$, $c=20$
f) $a=3$, $b=3$, $c=-18$
g) $a=1$, $b=12$, $c=36$
h) $a=9$, $b=-54$, $c=81$
Question 4
Solve the following using the quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$
a) $x^2+7x+12=0$
b) $x^2+10x+24=0$
c) $x^2+21x=0$
d) $x^2-8x+15=0$
e) $x^2-11x+28=0$
f) $x^2-16=0$
g) $x^2-3x-10=0$
h) $x^2-14x-15=0$
Question 5
Solve the following using the quadratic formula
a) $2x^2-10x+12=0$
b) $3x^2-3x-90=0$
c) $10x^2-60x+80=0$
d) $8x^2-72=0$
e) $8x^2-8x-240=0$
f) $5x^2-25x=0$
g) $42x^2-168x-1344=0$
h) $84x^2-168x-1260=0$
Question 6
Solve the following using the quadratic formula
a) $2x^2-5x-3=0$
b) $3x^2-16x+5=0$
c) $5x^2+36x+7=0$
d) $8x^2-14x+3=0$
e) $20x^2+13x-21=0$
f) $25x^2-9=0$
g) $6x^2+11x-7=0$
h) $45x^2-10x=0$
Question 7
Solve the following using the quadratic formula
a) $9x^2-12x+4=0$
b) $25x^2+80x+64=0$
c) $4x^2-25=0$
d) $3x+7=0$
e) $15y^2-17y-110=0$
f) $25z^2-441=0$
g) $8a-14=0$
h) $16q^2-120q+225$
Question 8
Solve the following using the quadratic formula. Leave answers in exact form
a) $x^2+3x+1=0$
b) $x^2+5x-1=0$
c) $x^2+x-9=0$
d) $x^2-x-4=0$
e) $2x^2+3x-7=0$
f) $3x^2-5x-3=0$
g) $2x^2+x-11=0$
h) $-2x^2+x+8=0$
Question 9
Solve the following using the quadratic formula. Leave answers as decimals to three decimal places
a) $x^2+7x-2=0$
b) $x^2-8x-11=0$
c) $x^2+x-13=0$
d) $4x^2+2x-5=0$
e) $2x^2-6x-1=0$
f) $3x^2-8x+2=0$
g) $4x^2+10x+1=0$
h) $6x^2-x-8=0$
Question 10
Solve the following using the quadratic formula. Leave answers in exact form, you need to simplify surds
a) $x^2+4x+2=0$
b) $x^2+6x+3=0$
c) $x^2-8x-2=0$
d) $2x^2-6x-5=0$
e) $4x^2-8x+2=0$
f) $7x^2+x-2=0$
g) $5x^2-10x+4=0$
h) $-12x^2+12x+5=0$
Question 11
Solve the following using the quadratic formula. Leave answers in exact form, you may need to simplify surds
a) $2x^2-7x-13=-10$
b) $2x^2-36=x$
c) $x^2-6x+7=32-3x^2-6x$
d) $6y^2=94$
e) $\frac{1}{3}x^2+x-\frac{1}{3}=0$
f) $0.4x^2+0.6x-0.5=0$
g) $5x^2+\frac{1}{2}x-0.2=0$
h) $\frac{1}{3}x^2+x-5=0.3x-2x^2-\frac{1}{8}$
Question 12
Answer the following
a) The solutions to a quadratic equation using the quadratic formula are $x=\frac{9+\sqrt{61}}{2}$ or $x=\frac{9-\sqrt{61}}{2}$. What was the quadratic equation?
b) The solutions to a quadratic equation using the quadratic formula are $x=\frac{\sqrt{3}-3}{3},\:x=-\frac{3+\sqrt{3}}{3}$. What was the quadratic equation?
c) A rectangular room is $3$m longer than it is wide. The area of the lawn is $10$m$^2$. What are the dimensions of the room?
d) The product of two consecutive numbers is $4556$. What are the two numbers?
e) Sam thinks of a number. He doubles it and adds 4 to it. He then multiplies this by his original number and gets $96$. What was his original number?
Question 13
Follow these steps to prove the quadratic formula
a) The general equation of a quadratic is $ax^2+bx+c=0$. Divide everything by $a$
b) What do you need to add to $x^2+\frac ba x$ to make the expression a perfect square?
c) Rewrite $x^2+\frac bax+\frac ca=0$ with the addition and subtraction of your answer to part b
d) Complete the square on your equation in part c
e) Collect everything not in brackets to the right hand side of the equation
f) Put everything on the right hand side of the equation into one fraction
g) Square root everything. Remember there will be a positive and a negative!
h) Make $x$ the subject. Everything on the right hand side should be in one fraction.
Answers
Question 1
a) $a=2$, $b=3$, $c=4$
b) $a=3$, $b=5$, $c=8$
c) $a=1$, $b=2$, $c=5$
d) $a=4$, $b=-5$, $c=-8$
e) $a=-1$, $b=-1$, $c=3$
f) $a=2$, $b=0$, $c=6$
g) $a=1$, $b=-3$, $c=0$
h) $a=0$, $b=1$, $c=2$
Question 2
a) $a=4$, $b=9$, $c=-2$
b) $a=1$, $b=-8$, $c=-3$
c) $a=1$, $b=0$, $c=2$
d) $a=22$, $b=0$, $c=-11$
e) $a=9$, $b=-5$, $c=1$
f) $a=73$, $b=-16$, $c=-3$
g) $a=20$, $b=-2$, $c=-21$
h) $a=201$, $b=181$, $c=-198$
Question 3
a) $x=-9$ or $x=-1$
b) $x=-1$ or $x=-2$
c) $x=-2$ or $x=-3$
d) $x=2$ or $x=3$
e) $x=2$ or $x=5$
f) $x=3$ or $x=-2$
g) $x=-6$
h) $x=3$
Question 4
a) $x=-3$ or $x=-4$
b) $x=-4$ or $x=-6$
c) $x=0$ or $x=-21$
d) $x=3$ or $x=5$
e) $x=4$ or $x=7$
f) $x=4$ or $x=-4$
g) $x=5$ or $x=-2$
h) $x=15$ or $x=-1$
Question 5
a) $x=2$ or $x=3$
b) $x=6$ or $x=-5$
c) $x=2$ or $x=4$
d) $x=3$ or $x=-3$
e) $x=6$ or $x=-5$
f) $x=0$ or $x=5$
g) $x=8$ or $x=-4$
h) $x=5$ or $x=-3$
Question 6
a) $x=3$ or $x=-\frac12$
b) $x=5$ or $x=\frac13$
c) $x=-\frac15$ or $x=-7$
d) $x=\frac32$ or $x=\frac14$
e) $x=\frac34$ or $x=-\frac75$
f) $x=\frac35$ or $x=-\frac35$
g) $x=\frac12$ or $x=-\frac73$
h) $x=0$ or $x=\frac29$
Question 7
a) $x=\frac23$
b) $x=-\frac85$
c) $x=\pm\frac52$
d) $x=-\frac73$
This is not a quadratic, so the quadratic formula fails
e) $y=\frac{10}{3}$ or $y=-\frac{11}{5}$
f) $z=\pm\frac{21}{5}$
g) $a=\frac74$
h) $q=\frac{15}{4}$
Question 8
a) $x=\frac{-3+\sqrt{5}}{2},\:x=\frac{-3-\sqrt{5}}{2}$
b) $x=\frac{-5+\sqrt{29}}{2},\:x=\frac{-5-\sqrt{29}}{2}$
c) $x=\frac{-1+\sqrt{37}}{2},\:x=\frac{-1-\sqrt{37}}{2}$
d) $x=\frac{1+\sqrt{17}}{2},\:x=\frac{1-\sqrt{17}}{2}$
e) $x=\frac{-3+\sqrt{65}}{4},\:x=\frac{-3-\sqrt{65}}{4}$
f) $x=\frac{5\pm\sqrt{61}}{6}$
g) $x=\frac{-1\pm\sqrt{89}}{4}$
h) $x=-\frac{-1\pm\sqrt{65}}{4}$ or $x=\frac{1\pm\sqrt{65}}{4}$
Question 9
a) $x=0.275$ or $x=-7.275$
b) $x=9.196$ or $x=-1.196$
c) $x=3.140$ or $x=-4.140$
d) $x=0.896$ or $x=-1.396$
e) $x=3.158$ or $x=-0.158$
f) $x=2.387$ or $x=0.279$
g) $x=-0.104$ or $x=-2.396$
h) $x=1.241$ or $x=-1.074$
Question 10
a) $x=\sqrt{2}-2,\:x=-2-\sqrt{2}$
b) $x=\sqrt{6}-3,\:x=-3-\sqrt{6}$
c) $x=4\pm3\sqrt{2}$
d) $x=\frac{3+\sqrt{19}}{2},\:x=\frac{3-\sqrt{19}}{2}$
e) $x=\frac{2\pm\sqrt{2}}{2}$
f) $x=\frac{-1\pm\sqrt{57}}{14}$
g) $x=\frac{5+\sqrt{5}}{5},\:x=\frac{5-\sqrt{5}}{5}$
h) $x=\frac{3\pm2\sqrt{6}}{6}$
Question 11
a) $x=\frac{7\pm\sqrt{73}}{4}$
b) $x=\frac{9}{2},\:x=-4$
c) $x=\pm\frac{5}{2}$
d) $y=\pm\sqrt{\frac{47}{3}}$
e) $x=\frac{-3+\sqrt{13}}{2},\:x=\frac{-3-\sqrt{13}}{2}$
f) $x=\frac{-3\pm\sqrt{29}}{4}$
g) $x=\frac{-1\pm\sqrt{17}}{20}$
h) $x=\frac{3\left(-7\pm3\sqrt{511}\right)}{140}$
Question 12
a) $x^2-9x+5=0$
b) $3x^2+6x+2=0$
c) $2$m by $5$m
d) $67$ and $68$
e) His original number was $6$
Question 13
a) $x^2+\frac bax+\frac ca=0$
b) Add $\frac{b^2}{4a^2}$ to make $x^2+\frac bax+\frac{b^2}{4a^2}$
c) $x^2+\frac bax+\frac{b^2}{4a^2}-\frac{b^2}{4a^2}+\frac ca=0$
d) $\left(x+\frac{b}{2a}\right)^2-\frac{b^2}{4a^2}+\frac ca=0$
e) $\left(x+\frac{b}{2a}\right)^2=\frac{b^2}{4a^2}-\frac ca$
f) $\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}$
g) $x+\frac{b}{2a}={\frac{\pm\sqrt{b^2-4ac}}{2a}}$
h) $x={\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$
Solutions
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