08Ddiscriminant

=** The Discriminant **=

When solving a quadratic equation in the form: ... ... ... ... ax 2 + bx + c = 0
 * Recall **

We can use the Quadratic Formula:

The Discriminant
The ** discriminant ** is the name given to the expression __inside__ the square root sign in the Quadratic Formula.
 * The symbol for the discriminant is Δ (delta - Greek Capital D)
 * Δ = b 2 – 4ac

We have previously noted that when solving a quadratic equation there could be:
 * two real solutions
 * one real solution
 * no real solutions

The __number__ of solutions depends on the value of the discriminant.
 * If the discriminant is ** positive ** (Δ > 0), there are ** two real solutions **
 * If Δ is a** perfect square **, the solutions are ** rational **.
 * If the discriminant is ** zero ** (Δ = 0), there is ** one ** ** rational ** solution
 * If the discriminant is ** negative ** (Δ < 0), there are ** no real solutions **.

The reasons for these rules comes directly from the quadratic formula.

math \\ . \qquad \bullet \; \text{ If } \Delta > 0 \text{ the } \pm \text{ symbol comes into effect giving 2 real solutions} \\ . \qquad \qquad \circ \; \text{ If } \Delta \text{ is a perfect square, the square root part gives a rational result} \\ . \qquad \qquad \circ \; \text{ If } \Delta \text{ is } \underline{not} \text{ a perfect square, the two solutions will be irrational} math . math \\ . \qquad \bullet \; \text{ If } \Delta = 0 \text{ the square root part disappears and the one rational solution is } x = \dfrac{-b}{2a} \\ . \qquad \bullet \; \text{ If } \Delta < 0 \text{ the square root part is undefined, so no real solutions} math


 * Example 1 **

Use the discriminant to state the number and type of solutions for these quadratic equations
 * 1) 2x 2 + 4x + 10 = 0
 * 2) x 2 – 6x + 9 = 0
 * 3) 3x 2 + 5x – 4 = 0


 * Solution: **

1. .. 2x 2 + 4x + 10 = 0

math . \qquad a = 2 \quad b = 4 \quad c = 10 \\. \\ . \qquad \Delta = 4^2 - 4 \times 2 \times 10 \\ . \qquad \; \; \; = 16 - 80 \\ . \qquad \; \; \; = -64 \\. \\ . \qquad \Delta < 0 \; \text{ Hence no real solutions} math

2. .. x 2 – 6x + 9 = 0

math . \qquad a = 1 \quad b = -6 \quad c = 9 \\. \\ . \qquad \Delta = (-6)^2 - 4 \times 1 \times 9 \\ . \qquad \; \; \; = 36 - 36 \\ . \qquad \; \; \; = 0 \\. \\ . \qquad \Delta = 0 \; \text{ Hence one rational solution} math

3. .. 3x 2 + 5x – 4 = 0

math . \qquad a = 3 \quad b = 5 \quad c = -4 \\. \\ . \qquad \Delta = 5^2 - 4 \times 3 \times (-4) \\ . \qquad \; \; \; = 25 + 48 \\ . \qquad \; \; \; = 73 \\. \\ . \qquad \Delta > 0 \; \text{(and not a perfect square) Hence two irrational solutions} math

The Discriminant and Graphs
Remember that the solutions to the quadratic equation ... ... ax 2 + bx + c = 0

give the x-intercepts of the parabola ... ... y = ax 2 + bx + c

Hence the discriminant can give us information about the parabola.

math \\ . \qquad \bullet \; \text{ If } \Delta > 0 \text{ we have 2 x-intercepts} \\ . \qquad \bullet \; \text{ If } \Delta = 0 \text{ we have 1 x-intercept so the turning point is on the x-axis at } x = \dfrac{-b}{2a} \\ . \qquad \bullet \; \text{ If } \Delta < 0 \text{ we have no x-intercepts so the parabola is entirely above (or entirely below) the x-axis} math


 * Advanced Questions with Discriminant **


 * Example **

... ... For what value of k does x 2 + kx + 3 = 0 have: ... ... ... (i) two real solutions ... ... ... (ii) one real solution ... ... ... (iii) no real solutions


 * Solution:**

... ... Find the discriminant:

math . \qquad \Delta = b^2 - 4ac \\. \\ . \qquad \Delta = k^2 - 4 \times 1 \times 3 \\. \\ . \qquad \Delta = k^2 - 12 math

... ... ** (i) ** .. There will be two real solutions where:

math . \qquad \Delta > 0 \\. \\ . \qquad k^2 - 12 > 0 \\. \\ . \qquad k^2 > 12 \\. \\ . \qquad \big| k \big| > \sqrt{12} \\. \\ . \qquad \big| k \big| > 2 \sqrt{3} \\. \\ . \qquad k > 2\sqrt{3} \quad \text{ OR } \quad k < -2\sqrt{3} math

... ... ** (ii) ** .. There will be one real solution where:

math . \qquad \Delta = 0 \\. \\ . \qquad k^2 - 12 = 0 \\. \\ . \qquad k^2 = 12 \\. \\ . \qquad \big| k \big| = \sqrt{12} \\. \\ . \qquad k = \pm 2 \sqrt{3} math

... ... ** (iii) ** .. There will be no real solutions where:

math . \qquad \Delta < 0 \\. \\ . \qquad k^2 - 12 < 0 \\. \\ . \qquad k^2 < 12 \\. \\ . \qquad \big| k \big| < \sqrt{12} \\. \\ . \qquad \big| k \big| < 2 \sqrt{3} \\. \\ . \qquad -2\sqrt{3} < k < 2\sqrt{3} math

.