1. THE DISCRIMINANT

2. The roots of the quadratic equation ax 2 + bx + c = 0 are given by the formula: b 2 – 4ac is known as the discriminant. If b 2 – 4ac is positive, we find the square root of a positive number, which has two solutions. i.e. the quadratic has two real roots. If b 2 – 4ac is zero, the roots will be:i.e. the same. If b 2 – 4ac is negative, we have the square root of a negative number, which has no real solutions. i.e. the quadratic has no real roots.

3. There are three cases: b 2 – 4ac < 0 b 2 – 4ac > 0 b 2 – 4ac = 0 The quadratic has 2 distinct real roots. The quadratic has equal, or, repeated roots. The quadratic has no real roots.

4. Example 1: Find the values of k for which the equation x 2 + kx + k + 3 = 0 has repeated roots. The coefficients of the quadratic give:a = 1,b = k,c = k + 3 The condition for repeated roots is b 2 – 4ac = 0 k 2 – 4(1)(k + 3) = 0 k 2 – 4k – 12 = 0 ( k – 6 )( k + 2 ) = 0 k = 6 or –2 To illustrate the point, if k = 6, the quadratic is: x 2 + 6x + 9 = 0 which factorises as: ( x + 3 )( x + 3 ) = 0 and has rootsx = –3 ( twice ).

5. Example 2: Find the values of p for which the equation 4x 2 + ( p – 5 ) x + p = 0 has two distinct real roots. The coefficients of the quadratic give:a = 4,b = p – 5, c = p The condition for 2 distinct real roots is b 2 – 4ac > 0 (p – 5) 2 – 4(4)(p) > 0 p 2 – 10p + 25 – 16p > 0 p 2 – 26p + 25 > 0 ( p – 25 )( p – 1 ) > 0 A quadratic inequality, so using a sketch: Now we want the values of x where the quadratic is more than zero. i.e. p < 1, orp > 25 25 1

6. Example 3: Find the values of m for which the equation x 2 – mx + m + 2 = 0 has no real roots. The coefficients of the quadratic give:a = 1,b = – m,c = m + 2 The condition for no real roots is b 2 – 4ac < 0 (– m) 2 – 4(1)(m + 2) < 0 m 2 – 4m – 8 < 0 A quadratic inequality, so using a sketch: Now we want the values of x where the quadratic is less than zero. This does not factorise, so use the formula to find the roots: < m < i.e.

7. Summary of key points: This PowerPoint produced by R.Collins ; Updated Apr. 2009 For the quadratic equation ax 2 + bx + c = 0 If b 2 – 4ac > 0 the equation will have two distinct real roots. If b 2 – 4ac < 0 If b 2 – 4ac = 0 the equation will have no real roots. the equation will have repeated roots.