Quadratic Formula

The quadratic formula solves a quadratic equation in standard form:

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

Use it for equations of the form $ax^2 + bx + c = 0$ with $a \ne 0$. If a quadratic factors quickly, factoring is often faster. If it does not, the quadratic formula is the reliable method that still works.

What The Quadratic Formula Tells You

The formula gives the value or values of $x$ that make the quadratic equal to zero. In $ax^2 + bx + c = 0$, the numbers $a$, $b$, and $c$ are the coefficients you substitute into the formula.

The part under the square root,

$$b^2 - 4ac$$

is called the discriminant. It helps you predict the kind of answer before you finish the arithmetic:

1. If $b^2 - 4ac > 0$, there are two distinct real solutions.
2. If $b^2 - 4ac = 0$, there is one repeated real solution.
3. If $b^2 - 4ac < 0$, there are no real solutions. In that case, the solutions are complex.

That quick check is useful because it tells you what to expect from the formula.

Why It Works

A quadratic can have up to two $x$-values where its graph crosses the $x$-axis. The quadratic formula is the general result of completing the square, so it gives those intercepts directly without guessing factors.

You do not need to re-derive it every time. In practice, the main job is identifying $a$, $b$, and $c$ correctly and keeping the signs straight.

Worked Example: Solve $2x^2 + 3x - 2 = 0$

First identify the coefficients:

$$a = 2, \quad b = 3, \quad c = -2$$

Now substitute:

$$x = \frac{-3 \pm \sqrt{3^2 - 4(2)(-2)}}{2(2)}$$

Work inside the square root first:

$$3^2 - 4(2)(-2) = 9 + 16 = 25$$

So the formula becomes

$$x = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}$$

Now evaluate both branches:

$$x = \frac{-3 + 5}{4} = \frac{2}{4} = \frac{1}{2}$$ $$x = \frac{-3 - 5}{4} = \frac{-8}{4} = -2$$

So the solutions are

$$x = \frac{1}{2} \quad \text{and} \quad x = -2$$

You can check one root by substitution. When $x = \frac{1}{2}$,

$$2\left(\frac{1}{2}\right)^2 + 3\left(\frac{1}{2}\right) - 2 = \frac{1}{2} + \frac{3}{2} - 2 = 0$$

That confirms the value works.

Common Mistakes With The Quadratic Formula

1. Not rewriting the equation as $ax^2 + bx + c = 0$ first. If the right side is not zero, the coefficients are not ready for the formula.
2. Losing the sign of $b$ or $c$. If $b = -7$, then $-b = 7$, not $-7$.
3. Forgetting that the denominator is the whole $2a$. The entire numerator $-b \pm \sqrt{b^2 - 4ac}$ sits over $2a$.
4. Calculating only one case. The $\pm$ means you must check both the plus and minus versions.
5. Making arithmetic errors inside the discriminant. Small sign mistakes there change the entire answer.

When To Use The Quadratic Formula

The quadratic formula is most useful when:

1. A quadratic does not factor cleanly.
2. You want a method that always works for standard-form quadratics.
3. You want to know how many real solutions to expect from the discriminant.
4. You are comparing methods such as factoring, completing the square, and graphing.

Try A Similar Problem

Solve $x^2 - 6x + 5 = 0$ with the same steps: identify $a$, $b$, and $c$, compute the discriminant, and evaluate both branches. If you want a useful comparison, factor it afterward and check that both methods give the same roots.