Ronald Hui Tak Sun Secondary School Mathematics Ronald Hui Tak Sun Secondary School

2016 HKDSE Ronald HUI

2017 HKDSE Ronald HUI

Follow-up question Factorize x2 + 2x – 15. Solution List all the possible pairs of factors of –15. Check each pair of factors by the cross-method as follows:

Quadratic Equations in One Unknown

highest degree of the unknown is 2 What is quadratic equation in one unknown? x 2  5x + 6 = 0, 2 8x 2 + 2x  3 = 0, 2 highest degree of the unknown is 2 one unknown x 4y 2 = y + 1, 2 5y 2 = 20 2 one unknown y These equations are all quadratic equations in one unknown.

a, b and c are real numbers General form of quadratic equations in one unknown ax2 + bx + c = 0 a  0 a, b and c are real numbers If a = 0, the equation becomes a linear equation bx + c = 0. Rearranging terms For example: (i) 2x2  5x + 6 = 0 2  5 + 6 (ii) 2x  3 + x2 = 0 We usually keep a positive. x2 + 2x  3 = 0 1 + 2  3 a b c a b c

In fact, all quadratic equations can be written in general form. Rewrite the following quadratic equations in general form. By transposing terms, 2x2 + 5x  3 = 0 2x2 + 5x = 3 By expanding the equation, x2 + x + 3 = 0 x(x + 1) + 3 = 0 By expanding and transposing terms, x2  4x + 3 = 0 (x  2)2 = 1

What is a root of a quadratic equation ax2 + bx + c = 0? Roots of a quadratic equation What is a root of a quadratic equation ax2 + bx + c = 0? A root of an equation is a value of x that satisfies the equation. For example, 3 is a root of the equation x2  9 = 0.  32 – 9 = 0

Is 1 a root of the equation x 2  5x  6 = 0? Follow-up question Is 1 a root of the equation x 2  5x  6 = 0? Substitute x = 1 into the equation. L.H.S. = (1)2  5(1)  6 = 0 = R.H.S.  1 is a root of the equation x 2  5x  6 = 0.

Solving Quadratic Equations by Factor Method

For any expression in the form (px  r)(qx  s), For any two real numbers a and b, if (px  r)(qx  s) = 0 if ab = 0 a = 0 or b = 0 px  r = 0 or qx  s = 0

How to solve a quadratic equation using the factor method? ax2 + bx + c = 0 (px  r)(qx  s) = 0 factorize ax2 + bx + c px  r = 0 or qx  s = 0 x = or x = r p s q The roots of ax2 + bx + c = 0 are and . r p s q

x2 + x  12 = 0 using the factor method? Can you solve x2 + x  12 = 0 using the factor method? x2 + x  12 = 0 Factorize x2 + x  12 first. (x  3)(x + 4) = 0 x –3 +4 –3x +4x = +x x  3 = 0 or x + 4 = 0 x = 3 or x = 4  The roots of x2 + x  12 = 0 are 3 and 4.

 The roots of 2x2  x  6 = 0 are and 2. Follow-up question Solve 2x2  x  6 = 0 using the factor method. 2x2  x  6 = 0 Factorize 2x2 – x  6. (2x + 3)(x  2) = 0 2x + 3 = 0 or x  2 = 0 3 x =  2 or x = 2  The roots of 2x2  x  6 = 0 are and 2. 3 2