C2: Quadratic Functions and Discriminants Dr J Frost Last modified: 2 nd September 2013

Starter Solve the following: ? ? ? ? ? ?

Completing the Square ? ? ? ? ? ? Put the following in the form p(x+q) 2 + r

Completing the Square Put the following in the form p – q(x + r) 2 2x – 3 – x 2 -2 – (x-1) 2 7 – 6x – x 2 16 – (x+3) 2 5 – 2x 2 – 8x13 – 2(x+3) 2 18x + 10 – 3x 2 37 – 3(x-3) 2 ? ? ? ?

Exercises ? ? ? ? ? ? ? ? ? x – x 2 = 12 – (x-3) 2 10 – 8x – x 2 = 26 – (x+4) 2 10x – 8 – 5x 2 = -3 – 5(x-1) 2 1 – 36x – 6x 2 = 55 – 6(x+3)

Solving Equations by Completing the Square ? ?

Your go… ? ?

Examples Exercise 2D – Page 21 ? ? ? ? ? E2 Given that for all values of x: 3x x + 5 = p(x+q) 2 + r a)Find all the values of p, q and r. b)Hence solve the equation 3x x + 5 = 0 p = 3, q = 2, r = -7 x = -2  √ (7/3) ? ? E1

The Quadratic Formula ? Proof?

The Discriminant Roots ? What formula do we know to find these roots?

The Discriminant b 2 – 4ac is known as the discriminant.

The Discriminant x 2 + 3x + 4 EquationDiscriminantNumber of Roots -70 x 2 – 4x x 2 – 4x x 2 – 6x – x – 4 – 3x – x 2 42 ?? ?? ?? ?? ?? ??

The Discriminant y = ax 2 + bx + c x y x y x y b 2 – 4ac > 0b 2 – 4ac = 0b 2 – 4ac < 0 ?? ? What can we say about the discriminant in each case? 2 roots/solutions1 roots/solutions0 roots/solutions ???

The Discriminant a) p = 4 (reject p = -1) b) x = -4 ? ?

The Discriminant Find the values of k for which x 2 + kx + 9 = 0 has equal roots. k =  6 Find the values of k for which x 2 – kx + 4 = 0 has equal roots. k =  4 Find the values of k for which kx 2 + 8x + k = 0 has equal roots. k =  ? ? ? We’ll revisit this topic after we’ve done Inequalities. Find the values of k for which kx 2 + (2k+1)x = 4 has equal roots. k = -1  0.5 √ 3 4 ?

Sketching Quadratics Sketch y = x 2 + 2x + 1Sketch y = x 2 + x – 2 x y x y Sketch y = -x 2 + 2x + 3 x y 3 3 Sketch y = 2x 2 – 5x – 3 x y ?? ??

Sketching Quadratics Sketch y = x 2 – 4x + 5Sketch y = x 2 + 6x + 12 x y x y (2, 1) 5 (-3, 3) 12 Sketch y = -x 2 + 2x – 3 y -3 (1,-2) Sketch y = -2x 2 – 12x – 22 y -22 (-3, -4) ?? ?? All of the following have no roots. Complete the square in order to find the min/max point.

Exercises Sketch the following. Make sure you indicate any intersections with the axes. Q8-10 have no roots – complete the square in order to indicate the min/max point. y = x 2 – 9 y = x 2 – 3 y = 1 - x 2 y = x 2 + 2x – 35 y = 2x 2 + x – 3 y = 6 – 10x – 4x 2 y = 15x – 2x 2 y = x 2 – 10x + 28 y = x 2 + 8x + 19 y = 2x – 2 – x