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Solving Quadratic Equations by Factorisation A quadratic equation is an equation of the form a x 2 + b x + c = 0, a  0 One of the methods used to solve quadratic equations is by factoring. If the product of two numbers is 0 then one (or both) of the numbers must be 0. So if xy = 0 either x = 0 or y = 0 Considering some specific numbers: If 8 x x = 0 then x = 0 If y x 15 = 0 then y = 0 Intro

Ex1 and 2 Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0, a  0 The first step in solving is to rearrange them (if necessary) into the form shown above. x2 = 4x x2 = 4x Example 1: Solve 6 x 2 = – 9 x Example 2: Solve x 2 – 4 x = 0 x ( x – 4) = 0 either x = 0 or x – 4 = 0 if x – 4 = 0 then x = 4 Solutions (roots) are x = 0, x = 4 6 x 2 + 9 x = 0 3 x (2 x + 3) = 0 either 3 x = 0 or 2 x + 3 = 0  x = 0 or x = – 1½ rearrange factorise rearrange factorise

Ex 3 and 4 Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0, a  0 4x2 = 9 4x2 = 9 Example 3: Solve x 2 – x – 12 = 0 Example 4: Solve 4 x 2 – 9 = 0 (2 x + 3 ) (2 x – 3) = 0 ( Using the difference of 2 squares) rearrange factorise if 2 x + 3 = 0 then x = – 1½ if 2 x – 3 = 0 then x = 1½ Solutions (roots) are x = +/ – 1½ ( x + 3)( x – 4) = 0 if x + 3 = 0 then x = – 3 if x – 4 = 0 then x = 4 Solutions (roots) are x = – 3 or 4 The first step in solving is to rearrange them (if necessary) into the form shown above.

Ex 5 and 6 Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0, a  0 9x2 = 1 9x2 = 1 Example 5: Solve 6 x 2 = 3 – 7x Example 6: Solve 9 x 2 – 1 = 0 (3 x + 1 ) (3 x – 1) = 0 ( Using the difference of 2 squares) rearrange factorise if 3 x + 1 = 0 then x = – 1/3 if 3 x – 1 = 0 then x = 1/3 Solutions (roots) are x = +/ – 1/3 ( 2 x + 3)(3 x – 1) = 0 if 2 x + 3 = 0 then x = – 1½ if 3 x – 1 = 0 then x = 1/3 Solutions (roots) are x = – 1½ or 1/3 rearrange 6 x 2 + 7x – 3 = 0 The first step in solving is to rearrange them (if necessary) into the form shown above.

Questions Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0, a  0 Solve each of the following quadratic equations by factorisation. (a) 5 x 2 - 10 x=0 (b) 4 x 2 - 6 x = 0 x = 0 or 2 x = 0 or 1½ (c) x 2 + 3 x + 2 = 0 x = -1 or -2 (d) 4 x 2 - 9 = 0 x = +/- 1½ (e) 2 t 2 - 9 t - 5 = 0 x = -½ or 5 (f) 16 x 2 – 100 = 0 x = +/- 2½ (g) 5 x 2 = - 4 x 2 + 1 x = +/- 1/3 (h) x 2 - x = 0x = 3 or -3 (i) 12 x 2 - 13 x + 3 = 0 x = ¾ or 1/3