Solving quadratic equations

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

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

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

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 = 2.39 and x = 0.28

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?

By completing the square Solve 𝑥 2 −6𝑥+8=0 take the 8 to the other side of the = 𝑥 2 −6 𝑥 = −8 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 ) 2 =1 What is in the bracket goes in the bracket! This (𝑥 −3) 2 = 1 means put the -3 in the bracket (x – 3) = ± 1 square root and solve.

Equations involving fractions 𝑥 −3 12 𝑥 −7 =70 𝑥 x 12 𝑥 - 7 x 𝑥 -3 x 12 𝑥 - 3 x -7 = 70 12 – 7 𝑥 - 36 𝑥 + 21 = 70 33 – 7 𝑥 - 36 𝑥 = 70 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 −7 x = -4 or x = 9 7

Equations involving fractions 3 𝑥+2 + 1 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.