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C2: Quadratic Functions and Discriminants

C2: Quadratic Functions and Discriminants. Dr J Frost (jfrost@tiffin.kingston.sch.uk). Last modified: 2 nd September 2013. Starter. Solve the following:. ?. ?. ?. ?. ?. ?. Completing the Square. ?. Put the following in the form p( x+q ) 2 + r. ?. ?. ?. ?. ?.

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C2: Quadratic Functions and Discriminants

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  1. C2: Quadratic Functions and Discriminants Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 2nd September 2013

  2. Starter Solve the following: ? ? ? ? ? ?

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

  4. Completing the Square Put 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 ?

  5. Exercises 3 + 6x – x2 = 12 – (x-3)2 ? 10 1 ? 10 – 8x – x2 = 26 – (x+4)2 11 2 12 10x – 8 – 5x2 = -3 – 5(x-1)2 3 ? 4 ? 5 ? 6 ? ? 7 ? 8 1 – 36x – 6x2 = 55 – 6(x+3)2 13 ? 9

  6. Solving Equations by Completing the Square ? • Key Points: • No need to factorise the 2 out at the start, because . • Don’t forget the ?

  7. Your go… ? ?

  8. Examples Exercise 2D – Page 21 1 ? ? 3 ? 5 ? 7 ? 9 E1 Given that for all values of x: 3x2 + 12x + 5 = p(x+q)2 + r Find all the values of p, q and r. Hence solve the equation 3x2 + 12x + 5 = 0 E2 p = 3, q = 2, r = -7 ? ? x = -2 √(7/3)

  9. The Quadratic Formula ? Proof?

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

  11. The Discriminant • Looking at this formula, when do you think we only have: • No solutions for ? • One solution for ? • Two solutions for ? b2 – 4ac is known as the discriminant.

  12. The Discriminant Equation Discriminant Number of Roots x2 + 3x + 4 -7 ? ? 0 ? ? x2 – 4x + 1 12 2 ? ? 0 1 x2 – 4x + 4 ? ? 2 2x2 – 6x – 3 60 ? ? 0 x – 4 – 3x2 -47 ? ? 1 – x2 4 2

  13. The Discriminant y = ax2 + bx + c What can we say about the discriminant in each case? y y y x x x b2 – 4ac > 0 b2 – 4ac = 0 b2 – 4ac < 0 ? ? ? ? 2 roots/solutions ? 1 roots/solutions ? 0 roots/solutions

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

  15. The Discriminant 1 Find the values of k for which x2 + kx + 9 = 0 has equal roots. k =  6 ? 2 Find the values of k for which x2 – kx + 4 = 0 has equal roots. k =  4 ? 3 Find the values of k for which kx2 + 8x + k = 0 has equal roots. k =  4 ? 4 Find the values of k for which kx2 + (2k+1)x = 4 has equal roots. k = -1  0.5√3 ? We’ll revisit this topic after we’ve done Inequalities.

  16. Sketching Quadratics Sketch y = x2 + 2x + 1 Sketch y = x2 + x – 2 y ? y ? 1 x x -2 1 1 -2 Sketch y = -x2 + 2x + 3 Sketch y = 2x2 – 5x – 3 y y ? ? 3 x x -1 3 -0.5 3 -3

  17. All of the following have no roots. Complete the square in order to find the min/max point. Sketching Quadratics Sketch y = x2 – 4x + 5 Sketch y = x2 + 6x + 12 y ? y ? 5 12 (2, 1) (-3, 3) x x Sketch y = -x2 + 2x – 3 Sketch y = -2x2 – 12x – 22 y y ? ? (-3, -4) (1,-2) -3 -22

  18. 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 = x2 – 9 y = x2 – 3 y = 1 - x2 y = x2+ 2x – 35 y = 2x2 + x – 3 y = 6 – 10x – 4x2 y = 15x – 2x2 y = x2 – 10x + 28 y = x2 + 8x + 19 y = 2x – 2 – x2 1 2 3 4 5 6 7 8 9 10

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