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2. Solving Quadratic Equations by Factorisation A quadratic equation is an equation of the form a x 2 + b x + c = 0, a  0 The three methods used to solve quadratic equations are: 1.Factorisation 2.Completing the square 3.Using the common formula This presentation is an introduction to solving quadratic equations by factorisation. The following idea is used when solving quadratics by factorisation. 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

3. Ex1 and 2 Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0, a  0 Some quadratic equations can be solved by factorising and it is normal to try this method first before resorting to the other two methods discussed. 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

4. Ex 3 and 4 Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0, a  0 Some quadratic equations can be solved by factorising and it is normal to try this method first before resorting to the other two methods discussed. 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.

5. Ex 5 and 6 Solving Quadratic Equations by Factorisation a x 2 + b x + c = 0, a  0 Some quadratic equations can be solved by factorising and it is normal to try this method first before resorting to the other two methods discussed. 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.

6. 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 (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 x = +/- 2½ (g) 5 x 2 = - 4 x 2 + 1 x = +/- 1/3 (h) 2( x 2 + 5 x ) = - 12 x = -2 or -3 (i) 12 x 2 - 13 x + 3 = 0 x = ¾ or 1/3

7. Worksheet (a) 5 x 2 = 10 x (b) 4 x 2 - 6 x = 0 (c) x 2 + 3 x + 2 = 0 (d) 4 x 2 - 9 = 0 (e) 2 t 2 - 9 t - 5 = 0 (f) 16 x 2 = 100 (g) 5 x 2 = - 4 x 2 + 1 (h) 2( x 2 + 5 x ) = -12 (i) 12 x 2 - 13 x + 3 = 0 (a) 5 x 2 = 10 x (b) 4 x 2 - 6 x = 0 (c) x 2 + 3 x + 2 = 0 (d) 4 x 2 - 9 = 0 (e) 2 t 2 - 9 t - 5 = 0 (f) 16 x 2 = 100 (g) 5 x 2 = - 4 x 2 + 1 (h) 2( x 2 + 5 x ) = -12 (i) 12 x 2 - 13 x + 3 = 0 (a) 5 x 2 = 10 x (b) 4 x 2 - 6 x = 0 (c) x 2 + 3 x + 2 = 0 (d) 4 x 2 - 9 = 0 (e) 2 t 2 - 9 t - 5 = 0 (f) 16 x 2 = 100 (g) 5 x 2 = - 4 x 2 + 1 (h) 2( x 2 + 5 x ) = -12 (i) 12 x 2 - 13 x + 3 = 0 Worksheet