Solving Quadratic Equations

A quadratic equation is any equation that can be rearranged into the standard form:

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

Where $a$, $b$, and $c$ are numbers (coefficients) and $a \neq 0$.

1. The Quadratic Formula

When an equation cannot be factored easily, we use the Quadratic Formula. It works for every quadratic equation!

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

The part under the square root, $b^2 - 4ac$, is called the discriminant. It tells us how many solutions (roots) to expect:

If $b^2 - 4ac > 0$: Two distinct real roots (The graph crosses the x-axis twice).
If $b^2 - 4ac = 0$: One repeated root (The graph touches the x-axis once).
If $b^2 - 4ac < 0$: No real roots (The graph never crosses the x-axis).

Solve: $x^2 - 5x + 6 = 0$

Identify: $a = 1, b = -5, c = 6$

Step 1: Substitute into the formula:
$$x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(1)(6)}}{2(1)}$$

Step 2: Simplify:
$$x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm \sqrt{1}}{2}$$

Step 3: Final answers:
$x = \frac{5+1}{2} = 3$ and $x = \frac{5-1}{2} = 2$

[Visual: A U-shaped parabola crossing the x-axis at 2 and 3]

Solve: $x^2 - 4x + 4 = 0$

Identify: $a = 1, b = -4, c = 4$

Applying the formula:
$$x = \frac{4 \pm \sqrt{16 - 16}}{2} = \frac{4 \pm 0}{2}$$

Result: $x = 2$

[Visual: A U-shaped parabola whose vertex just touches the x-axis at 2]