Quadratics Formulas Completing the square x 2 bx

Quadratics

Formulas • Completing the square - x 2 + bx+ (b/2)2 = (x+b/2)2 • Quadratic Formula • Finding the discriminant -x= When: 0 > then equation has no real roots 0= then equation has 2 equal roots 0< then equation has 1 real root

Formulas • Finding the vertex - When equation in form y = A(x - h)2 + k, the vertex equals (h, k) • Solving inequalities - solve as you would a normal equation but note: division by negatives changes the direction of the inequality

Exam Tips • If you are unable to factorise, then use the quadratic formula • Watch your signs. Especially when using the quadratic formula.

Exam Style (1) The equation of a curve is y = 12 x - x 2 (i) Express 12 x - x 2 in the form a – (x+b) 2, stating the values of a and b (ii) state the maximum point of the curve y = 12 x - x 2 (iii) Find the set of values for x for which y ≥ 32.

Step by step guide 1. Using the completing the square formula [0 +] 12 x - x 2 = a – (x + b) = a - x 2 - 2 bx – b 2 = ( -2 b)x - x 2 + a - b 2 -2 b = 12 b = -6 a = b 2 a = 36 Therefore : 12 x - x 2 = 36 – (x-6) 2 2. We know that: y = 12 x - x 2 = - (x-6)2 +36 (answer from 1. rearranged) The maximum point is at (h, k). Therefore the maximum point is at (6, 36) 3. y ≥ 32 12 x - x 2 ≥ 32 0 ≥ x 2 – 12 x +32 0 ≥ (x - 4)(x - 8) Since it is below 0 (below the x axis) therefore: 4 ≤ x ≤ 8

Exam Style (2) When h(x) = x 2 – 8 x g(x) = 4 x – k Find the value of k such that h(x)= g(x) has only one solution.

Step by step guide 1. Put all terms to one side h(x) = g(x) X 2 - 8 x = 4 x – k [1]x 2 -12 x + k =0 2. Using the discriminant b 2 – 4 ac b 2 – 4 x 1 x k = 0 (for only one real solution) 144 – 4 x 1 x k = 0 4 k = 144 k = 36

Now it’s time to have a go yourself! DON’T LOOK AT THE ANSWERS UNTIL YOU’VE DONE THE QUESTIONS! Given g(x) = 2 x 2 – 16 x +25, (i) express g(x) in the form a(x+b)2 + c (ii)Write down the vertex of g(x) (iii) Determine the number of real solution.

Answers (i) g(x) = 2 x 2 – 16 x +25 = 2{x 2 – 8 x + 16} + 25 – 32 (perfect square) = 2(x-4)2 - 7 = 2(x-4)2 + -7 (ii) Using the above we can write the vertex: (4, -7) (iii) Using b 2 – 4 ac: (-16)2 – 4 x 25 256 – 200 56 ≥ 0 Therefore there are two real roots.