Training Quadratic and Polynomial Functions The Quadratic Formula and Discriminant
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The Quadratic Formula and Discriminant

22 min Quadratic and Polynomial Functions

The Quadratic Formula

The quadratic formula is the universal tool for solving any quadratic equation, even those that cannot be factored with integers. Derived by completing the square on the general form ax² + bx + c = 0, the formula gives both solutions in one compact expression.

The discriminant, the expression b² − 4ac under the square root, tells you the nature of the solutions before you even compute them: positive means two real solutions, zero means one repeated real solution, and negative means two complex solutions.

This lesson teaches you to apply the quadratic formula accurately, interpret the discriminant, and decide when factoring or the formula is the better choice.

Formula

For $ax^2 + bx + c = 0$:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The Discriminant

$\Delta = b^2 - 4ac$
$\Delta$Nature of Solutions
$\Delta > 0$Two distinct real solutions
$\Delta = 0$One repeated real solution
$\Delta < 0$Two complex (non-real) solutions
Example 1

$3x^2 - 2x - 5 = 0$

$a = 3$, $b = -2$, $c = -5$.

$\Delta = 4 + 60 = 64$. $\;x = \dfrac{2 \pm 8}{6}$.

$x = \dfrac{10}{6} = \dfrac{5}{3}$ or $x = \dfrac{-6}{6} = -1$.

Example 2

$x^2 + 4x + 7 = 0$

$\Delta = 16 - 28 = -12 < 0$. No real solutions.

Example 3

$x^2 - 6x + 9 = 0$

$\Delta = 36 - 36 = 0$. One repeated solution: $x = \dfrac{6}{2} = 3$.

Completing the Square

Method

To make $x^2 + bx$ a perfect square, add $\left(\dfrac{b}{2}\right)^2$ to both sides.

Example 4

$x^2 + 6x + 5 = 0$

  1. $x^2 + 6x = -5$
  2. Add $\left(\dfrac{6}{2}\right)^2 = 9$: $x^2 + 6x + 9 = 4$
  3. $(x + 3)^2 = 4$
  4. $x + 3 = \pm 2 \;\Rightarrow\; x = -1$ or $x = -5$
Example 5

$2x^2 - 8x + 1 = 0$

$a = 2$, $b = -8$, $c = 1$. $\Delta = 64 - 8 = 56$.

$x = \dfrac{8 \pm \sqrt{56}}{4} = \dfrac{8 \pm 2\sqrt{14}}{4} = \dfrac{4 \pm \sqrt{14}}{2}$.

Practice Problems

1. $x^2 - 4x - 5 = 0$ (formula)
2. $2x^2 + 3x - 2 = 0$ (formula)
3. Discriminant of $x^2 + 5x + 7 = 0$
4. $x^2 - 2x - 15 = 0$ (complete the square)
5. $x^2 + 8x + 12 = 0$ (formula)
6. $3x^2 - x - 4 = 0$
7. Discriminant of $4x^2 - 12x + 9 = 0$
8. $x^2 + 10x + 25 = 0$
9. $x^2 - 3x + 1 = 0$
10. $5x^2 + 2x - 1 = 0$
11. $x^2 + x + 1 = 0$ (classify solutions)
12. $x^2 - 4x + 1 = 0$ (complete the square)
Show Answer Key

1. $x = 5$ or $x = -1$

2. $x = \dfrac{1}{2}$ or $x = -2$

3. $\Delta = 25 - 28 = -3 < 0$; no real solutions

4. $(x-1)^2 = 16$; $x = 5$ or $x = -3$

5. $x = -2$ or $x = -6$

6. $x = \dfrac{4}{3}$ or $x = -1$

7. $\Delta = 144 - 144 = 0$; one repeated root $x = \dfrac{3}{2}$

8. $x = -5$ (repeated)

9. $x = \dfrac{3 \pm \sqrt{5}}{2}$

10. $x = \dfrac{-2 \pm \sqrt{24}}{10} = \dfrac{-1 \pm \sqrt{6}}{5}$

11. $\Delta = 1 - 4 = -3 < 0$; two complex solutions

12. $(x-2)^2 = 3$; $x = 2 \pm \sqrt{3}$

Solving Quadratics by Factoring Graphing Quadratics — Vertex Form