Solving Quadratic Equations

To solve a quadratic equation, rewrite it in standard form and find the value or values of $x$ that make the equation true. The standard form is

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

with $a \ne 0$. Most student problems come down to three methods: factoring, completing the square, or using the quadratic formula. The main skill is choosing the simplest method for the equation in front of you.

What Solving A Quadratic Means

You are looking for the roots, or solutions, of the equation. On a graph, these are the $x$-values where the parabola meets the $x$-axis.

A quadratic can have two real solutions, one repeated real solution, or no real solutions. If you are working over the complex numbers, every quadratic still has two solutions counting multiplicity.

How To Choose A Method

Before doing any algebra, move every term to one side so the other side is $0$. That makes the structure easier to see and helps you decide which method fits.

• If the expression factors cleanly, factoring is usually fastest.
• If the equation is close to a perfect square pattern, completing the square can be efficient.
• If neither approach is convenient, the quadratic formula works for any quadratic equation.

One more shortcut helps: the discriminant,

$$b^2 - 4ac$$

which tells you what kind of real solutions to expect.

• If $b^2 - 4ac > 0$, there are two distinct real solutions.
• If $b^2 - 4ac = 0$, there is one repeated real solution.
• If $b^2 - 4ac < 0$, there are no real solutions.

That does not solve the equation by itself, but it tells you what kind of answer should make sense before you start calculating.

The Three Main Methods

Factoring

Factoring works when the quadratic can be rewritten as a product such as

$$(x - 2)(x - 3) = 0$$

Then use the zero-product rule: if a product is $0$, at least one factor must be $0$. So the solutions are $x = 2$ and $x = 3$.

Completing The Square

Completing the square rewrites the quadratic in a form like

$$(x - h)^2 = k$$

This is especially useful when factoring is awkward and you want to see the equation as a squared expression.

Quadratic Formula

The quadratic formula always applies when the equation is in standard form:

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

It is the most reliable general method, but it is not always the fastest if the quadratic factors immediately.

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

This equation is already in standard form, so first check whether it factors. You need two numbers that multiply to $6$ and add to $-5$. Those numbers are $-2$ and $-3$, so

$$x^2 - 5x + 6 = (x - 2)(x - 3)$$

Now solve

$$(x - 2)(x - 3) = 0$$

Now set each factor equal to zero:

$$x - 2 = 0 \quad \text{or} \quad x - 3 = 0$$

So the solutions are

$$x = 2 \quad \text{or} \quad x = 3$$

Check both answers in the original equation:

$$2^2 - 5(2) + 6 = 4 - 10 + 6 = 0$$ $$3^2 - 5(3) + 6 = 9 - 15 + 6 = 0$$

Both checks work, so the solutions are correct.

Common Mistakes

One common mistake is choosing a method before moving all terms to one side. For example, solving $x^2 = 5x - 6$ becomes much easier after rewriting it as $x^2 - 5x + 6 = 0$.

Another mistake is dropping one solution. Quadratics can have two real solutions, so after factoring or using the $\pm$ in the quadratic formula, make sure you keep both branches when they exist.

A third mistake is using the quadratic formula with the wrong signs for $a$, $b$, or $c$. This usually happens when the equation is not first written in standard form.

Where You Use This

Quadratic equations appear in algebra, graphing, optimization, and motion problems. If a relationship includes a squared variable, solving a quadratic is often the step that gives the meaningful value of $x$.

The method depends on the equation. A clean factorable quadratic rewards pattern recognition. A messier one is often better handled by the quadratic formula.

Try A Similar Problem

Try solving $x^2 - 7x + 12 = 0$ and choose the method before you calculate. A useful next step is to solve the same equation by factoring and by the quadratic formula, then compare which method feels more direct.