1. AS Mathematics Algebra – Graphical solution of quadratic equations

2. Objectives Be able to factorise quadratic expressions Be able to sketch the graph of a quadratic equation Be able to solve quadratic equations by graphical methods

3. Quadratic equations When solving a quadratic equation you should always try to factorise it first. 3x 2 - 12x - 4x + 16 = 0 3x(x - 4) - 4(x - 4) = 0 sum = -16 product = 3(16) = 48 } -4 -12 (x - 4)(3x - 4) = 0 Have you checked your answer? 3x 2 - 16x + 16 = 0 Either x - 4 = 0 or 3x - 4 = 0 x = 4 or x = 4 / 3 The roots of this equation are x = 4 and x = 4 / 3 ___________ Recap: Example 1

4. What would the graph of y = 3x 2 – 16x + 16 look like? The roots of an equation tell us where the curve crosses the x- axis (the line y = 0) +ve coefficient of x 2 → The equation has 2 roots, x = 4 / 3 & x= 4 y = 3x 2 - 16x + 16

5. Example 2 Factorise & solve to find the roots 3x 2 - 6x + 3 = 0 3(x 2 - 2x + 1) = 0 3(x - 1)(x - 1) = 0 3(x - 1) 2 = 0 (x - 1) 2 = 0 x – 1 = 0 x = 1The equation has one root____ What does this tell us about the sketch of the graph? Find the roots of the quadratic equation3x 2 - 6x + 3 = 0 Use this information to draw a sketch graph of the curve y = 3x 2 - 6x + 3

6. The root of the equation is x = 1 → the curve only touches the x-axis at x = 1 +ve coefficient of x 2 → root x = 1 Notice the line of symmetry, x = 1 y = 3x 2 - 6x + 3

7. Example 3 a) Draw a graph of the curve y = 2x 2 + 3x - 2 b) Use your graph to find an approximate solution to the equation 2x 2 + 3x - 12 = 0 sum = 3, product = -42x 2 + 3x - 2 = 0 2x 2 - x + 4x - 2 = 0 x(2x - 1) + 2(2x - 1) = 0 (2x - 1)(x + 2) = 0 2x - 1 = 0 or x + 2 = 0 x = 1 / 2 or x = -2 The roots of the equation are x = ½ & x = -2 ____

8. +ve coefficient of x 2 → The roots of the equation are x = -2 & x = ½ → the curve cuts y = 0 at x = -2 & x = ½ root x = ½ root x = -2 y = 2x 2 + 3x - 2 Notice the line of symmetry, x = -1 1 / 2

9. b) Use your graph to find an approximate solution to the equation 2x 2 + 3x - 12 = 0 You need a fairly accurate graph to find the solutions of equations. Draw it by hand on graph paper, or use a graphing package / graphical calculator. Compare the quadratic we’ve just drawn, y = 2x 2 + 3x - 2 with 2x 2 + 3x - 12 = 0. Rearrange 2x 2 + 3x - 12 = 0 so that the LHS reads 2x 2 + 3x - 2 = ….. We get 2x 2 + 3x - 2 = 10 On the same graph as y = 2x 2 + 3x – 2 draw the graph of y = 10.

10. y = 2x 2 + 3x - 2 y = 10 Where the two lines cross y = 2x 2 + 3x - 2 = 0 AND y = 10. The graph shows 2 roots to the equation In other words 2x 2 + 3x - 2 = 10 or 2x 2 + 3x - 12 = 0 x = -3.3 & x = 1.8 (approximately)