1. Solving quadratic equations

2. Using the = 0 method x( 3x + 2)(x -1) = 0
x = x + 2 = 0 and x – 1 = 0
3x = x = 1
x = −2 3
Same thing for any number of brackets such as (x+2)(3x -4)( x + 1) = 0

3. Using the square root method
Solve
(2𝑥 −1) 2 = 16 square root both sides
(2x – 1) = ± remember to use the + or – sign
(2x – 1) = ± 4 find x for +4 and then for -4
(2x – 1) = and (2x – 1) = -4
2x = 5 2x = -3
x = x = −3 2

4. By factorising 2 𝑥 2 +11𝑥+12=0 open factors of 2 𝑥 2 and 12
(2x + 3)(x + 4) = x
2x + 3 = 0 and x + 4 = x cross multiply to see if
x = and x = they add up to 11

5. By using the equation 3 𝑥 2 - 8x + 2 = 0 where a = 3, b = -8 and c = 2
The equation is given in the equations page
𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎
𝑥= − −8 ± (−8) 2 −4(3)(2) 2 𝑥 3 use calculator to work it out twice, one for + and
one for –
x = and x = 0.28

6. Problem involving quadratics
Use the rectangle for find the value of x.
x(2x -1) = 30
2𝑥 2 −𝑥=30
2𝑥 2 −𝑥 −30=0
Use formula x = −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 to find the value of x
Think : one value would not be suitable. Which one is it?

7. By completing the square
Solve 𝑥 2 −6𝑥+8= take the 8 to the other side of the =
𝑥 2 −6 𝑥 = − take the number in front of x and divide it by 2
Square it and put it on either side of the =
𝑥 2 −6𝑥 + −3 2 = −8 + (−3) 2 On LHS, form a bracket with x
(𝑥 ) = What is in the bracket goes in the bracket! This
(𝑥 −3) = means put the -3 in the bracket
(x – 3) = ± square root and solve.

8. Equations involving fractions
𝑥 − 𝑥 −7 =70
𝑥 x 12 𝑥 - 7 x 𝑥 -3 x 12 𝑥 - 3 x -7 = 70
12 – 7 𝑥 𝑥 = 70
33 – 7 𝑥 𝑥 = take the 70 on the other side and group like terms
−37 −7𝑥 − 36 𝑥 =0 multiply by x all terms
−7𝑥 2 −37𝑥 −36=0 now use the equation
𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎
𝑥= 37± (−37) 2 −4(−7)(−36) 2𝑥 − x = or x = 9 7

9. Equations involving fractions
3 𝑥 𝑥+4 = 2 Use the lCM
3 𝑥+4 +(𝑥+2) (𝑥+2)(𝑥+4) = 2 simplify
4𝑥+14 (𝑥+2)(𝑥+4) = 2 cross multiply
4𝑥+14 = 2(𝑥+2)(𝑥+4)
4𝑥+14 = 2( 𝑥 2 +6𝑥+8)
4𝑥+14 =2𝑥 2 +12𝑥+16
0 = 2𝑥 2 +12𝑥 −4𝑥+16 −14
0= 2𝑥 2 +8𝑥+2 Solve by factorisation or using the equation.