C2: Quadratic Functions and Discriminants

Dr J Frost (jfrost@tiffin.kingston.sch.uk)

Last modified: 2nd September 2013

Starter

Solve the following:

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Completing the Square

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

Completing the SquarePut the following in the form p – q(x + r)2

2x – 3 – x2 -2 – (x-1)2

7 – 6x – x2 16 – (x+3)2

5 – 2x2 – 8x 13 – 2(x+3)2

18x + 10 – 3x2 37 – 3(x-3)2

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Exercises

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3 + 6x – x2 = 12 – (x-3)2

10 – 8x – x2 = 26 – (x+4)2

10x – 8 – 5x2 = -3 – 5(x-1)2

1 – 36x – 6x2 = 55 – 6(x+3)2

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Solving Equations by Completing the Square

Key Points:• No need to factorise the 2 out at the start, because .• Don’t forget the

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Your go…

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Examples

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Exercise 2D – Page 21

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E2 Given that for all values of x:3x2 + 12x + 5 = p(x+q)2 + ra) Find all the values of p, q and r.b) Hence solve the equation 3x2 + 12x + 5 = 0

p = 3, q = 2, r = -7

x = -2 √(7/3)

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E1

The Quadratic Formula

𝑎𝑥2+𝑏𝑥+𝑐=0

𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎

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Proof?

The Discriminant

𝑥

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Roots

𝑦=𝑎𝑥2+𝑏𝑥+𝑐

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What formula do we know to find these roots?

The Discriminant

Looking at this formula, when do you think we only have:• No solutions for ?• One solution for ?• Two solutions for ?

𝑥=−𝑏±√𝑏2−4𝑎𝑐2𝑎

b2 – 4ac is known as the discriminant.

The Discriminant

x2 + 3x + 4Equation Discriminant Number of Roots

-7 0

x2 – 4x + 1 12 2

x2 – 4x + 4 0 1

2x2 – 6x – 3 60 2

x – 4 – 3x2 -47 0

1 – x2 4 2

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The Discriminant

y = ax2 + bx + c

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b2 – 4ac > 0 b2 – 4ac = 0 b2 – 4ac < 0? ? ?

What can we say about the discriminant in each case?

2 roots/solutions 1 roots/solutions 0 roots/solutions? ? ?

The Discriminant

a) p = 4 (reject p = -1)

b) x = -4

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The Discriminant

Find the values of k for which x2 + kx + 9 = 0 has equal roots.

k = 6

Find the values of k for which x2 – kx + 4 = 0 has equal roots.

k = 4

Find the values of k for which kx2 + 8x + k = 0 has equal roots.

k = 4

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We’ll revisit this topic after we’ve done Inequalities.

Find the values of k for which kx2 + (2k+1)x = 4 has equal roots.

k = -1 0.5√3

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Sketching Quadratics

Sketch y = x2 + 2x + 1 Sketch y = x2 + x – 2

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-2 1-2

Sketch y = -x2 + 2x + 3

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-1 3

Sketch y = 2x2 – 5x – 3

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-0.5 3

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Sketching Quadratics

Sketch y = x2 – 4x + 5 Sketch y = x2 + 6x + 12

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(2, 1)5

(-3, 3)

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Sketch y = -x2 + 2x – 3 y

-3(1,-2)

Sketch y = -2x2 – 12x – 22 y

-22

(-3, -4)

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All of the following have no roots. Complete the square in order to find the min/max point.

ExercisesSketch 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 = x2 – 9y = x2 – 3y = 1 - x2

y = x2 + 2x – 35y = 2x2 + x – 3y = 6 – 10x – 4x2

y = 15x – 2x2

y = x2 – 10x + 28y = x2 + 8x + 19 y = 2x – 2 – x2

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