Quadratic Equations Points of intersection of linear graphs

Quadratic Equations Points of intersection of linear graphs an quadratic graphs

Quadratic Equations Finding solutions Find the solutions for y = x 2 – 2 x + 2 and y = 7 We draw the quadratic curve y = x 2 – 2 x + 2 and the line y=7 The solutions are where the quadratic curve crosses the straight line y = 7

Quadratic Equations y = x 2 - 2 x + 2 Solutions are 11 10 x = -1. 45 and x = 3. 45 9 8 y=7 7 6 5 4 3 2 1 0 -2 -1 0 1 2 3 4

Quadratic Equations Now review the solutions to x 2 – 2 x – 5 = 0 Solutions are x = -1. 45 and x = 3. 45 Is this a coincidence?

Quadratic Equations The solutions for y = x 2 – 2 x + 2 and y = 7 are where x 2 – 2 x + 2 = 7 Rearrange the equation by subtracting 7 from both sides x 2 – 2 x – 5 = 0

10 Quadratic Equations 9 8 y = x 2 – 2 x + 2 7 6 5 4 -7 3 2 1 0 -2 -1 0 1 22 33 44 -1 -2 -3 -4 -5 -6 y = x 2 – 2 x - 5 If we rearrange the equation we find that we have the same solutions because the process of rearranging translates (moves) the graph

Quadratic Equations This applies to the intersections of other lines Show where the curve of y = x 2 + 2 x – 4 crosses the line y = x - 3 Solutions are x = -1. 6 x = 0. 6

Quadratic Equations The solutions for y = x 2 – 2 x - 4 and y = x - 3 are where x 2 – 2 x - 4 = x - 3 Rearrange the equation - x from both sides + 3 to both sides x 2 – 3 x - 4 = - 3 x 2 – 3 x - 1 = 0 The solutions to this will be the same as the solutions to x 2 – 2 x - 4 = x - 3