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Algebra - Quadratic Formula

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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) $2x^2+3x+4=0$  a s v d
b) $3x^2+5x+8=0$  a s v d
c) $x^2+2x+5=0$  a s v d
d) $4x^2-5x-8=0$  a s v d
e) $-x^2-x+3=0$  a s v d
f) $2x^2+6=0$  a s v d
g) $x^2-3x=0$  a s v d
h) $x+2=0$  a s v d


Question 2


Find $a$, $b$ and $c$ by rearranging into the form $ax^2+bx+c=0$

a) $4x^2+9x=2$  a s v d
b) $x^2=8x+3$  a s v d
c) $x^2+6x=6x-2$  a s v d
d) $24x^2-3=2(x^2+4)$  a s v d
e) $3x^2+1=\frac{5x+2}{3}$  a s v d
f) $4(9x-2)=\frac{3-x^2}{2x}$  a s v d
g) $x^2(x-4)+3=x^3-\frac{6+2x}{5}$  a s v d
h) $(2x+3)(9x-5)-3=\frac{x(6-3x)}{11}$  a s v d


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 s v d
b) $a=1$, $b=3$, $c=2$  a s v d
c) $a=1$, $b=5$, $c=6$  a s v d
d) $a=1$, $b=-5$, $c=6$  a s v d
e) $a=2$, $b=-14$, $c=20$  a s v d
f) $a=3$, $b=3$, $c=-18$  a s v d
g) $a=1$, $b=12$, $c=36$  a s v d
h) $a=9$, $b=-54$, $c=81$  a s v d


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 s v d
b) $x^2+10x+24=0$  a s v d
c) $x^2+21x=0$  a s v d
d) $x^2-8x+15=0$  a s v d
e) $x^2-11x+28=0$  a s v d
f) $x^2-16=0$  a s v d
g) $x^2-3x-10=0$  a s v d
h) $x^2-14x-15=0$  a s v d


Question 5


Solve the following using the quadratic formula

a) $2x^2-10x+12=0$  a s v d
b) $3x^2-3x-90=0$  a s v d
c) $10x^2-60x+80=0$  a s v d
d) $8x^2-72=0$  a s v d
e) $8x^2-8x-240=0$  a s v d
f) $5x^2-25x=0$  a s v d
g) $42x^2-168x-1344=0$  a s v d
h) $84x^2-168x-1260=0$  a s v d


Question 6


Solve the following using the quadratic formula

a) $2x^2-5x-3=0$  a s v d
b) $3x^2-16x+5=0$  a s v d
c) $5x^2+36x+7=0$  a s v d
d) $8x^2-14x+3=0$  a s v d
e) $20x^2+13x-21=0$  a s v d
f) $25x^2-9=0$  a s v d
g) $6x^2+11x-7=0$  a s v d
h) $45x^2-10x=0$  a s v d


Question 7


Solve the following using the quadratic formula

a) $9x^2-12x+4=0$  a s v d
b) $25x^2+80x+64=0$  a s v d
c) $4x^2-25=0$  a s v d
d) $3x+7=0$  a s v d
e) $15y^2-17y-110=0$  a s v d
f) $25z^2-441=0$  a s v d
g) $8a-14=0$  a s v d
h) $16q^2-120q+225$  a s v d


Question 8


Solve the following using the quadratic formula. Leave answers in exact form

a) $x^2+3x+1=0$  a s v d
b) $x^2+5x-1=0$  a s v d
c) $x^2+x-9=0$  a s v d
d) $x^2-x-4=0$  a s v d
e) $2x^2+3x-7=0$  a s v d
f) $3x^2-5x-3=0$  a s v d
g) $2x^2+x-11=0$  a s v d
h) $-2x^2+x+8=0$  a s v d


Question 9


Solve the following using the quadratic formula. Leave answers as decimals to three decimal places

a) $x^2+7x-2=0$  a s v d
b) $x^2-8x-11=0$  a s v d
c) $x^2+x-13=0$  a s v d
d) $4x^2+2x-5=0$  a s v d
e) $2x^2-6x-1=0$  a s v d
f) $3x^2-8x+2=0$  a s v d
g) $4x^2+10x+1=0$  a s v d
h) $6x^2-x-8=0$  a s v d


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 s v d
b) $x^2+6x+3=0$  a s v d
c) $x^2-8x-2=0$  a s v d
d) $2x^2-6x-5=0$  a s v d
e) $4x^2-8x+2=0$  a s v d
f) $7x^2+x-2=0$  a s v d
g) $5x^2-10x+4=0$  a s v d
h) $-12x^2+12x+5=0$  a s v d


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 s v d
b) $2x^2-36=x$  a s v d
c) $x^2-6x+7=32-3x^2-6x$  a s v d
d) $6y^2=94$  a s v d
e) $\frac{1}{3}x^2+x-\frac{1}{3}=0$  a s v d
f) $0.4x^2+0.6x-0.5=0$  a s v d
g) $5x^2+\frac{1}{2}x-0.2=0$  a s v d
h) $\frac{1}{3}x^2+x-5=0.3x-2x^2-\frac{1}{8}$  a s v d


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 s v d

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 s v d

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 s v d

d) The product of two consecutive numbers is $4556$. What are the two numbers?  a s v d

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 s v d



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 s v d

b) What do you need to add to $x^2+\frac ba x$ to make the expression a perfect square?  a s v d

c) Rewrite $x^2+\frac bax+\frac ca=0$ with the addition and subtraction of your answer to part b  a s v d

d) Complete the square on your equation in part c  a s v d

e) Collect everything not in brackets to the right hand side of the equation  a s v d

f) Put everything on the right hand side of the equation into one fraction  a s v d

g) Square root everything. Remember there will be a positive and a negative!  a s v d

h) Make $x$ the subject. Everything on the right hand side should be in one fraction.  a s v d