How to Solve a Quadratic Equation? (+FREE Worksheet!)

Algebra 1

How to Solve a Quadratic Equation

Solving a quadratic equation \(ax^2 + bx + c = 0\) means finding the x-values that make it true — its roots. You can factor, use the square root for simple cases, or fall back on the quadratic formula when nothing factors. We’ll cover each path, with a solver and a worksheet maker a tap away.

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Illustration of students learning How to Solve a Quadratic Equation

Solving a quadratic equation — \(ax^2 + bx + c = 0\) — means finding the x-values that make it true, called its roots or solutions. A quadratic can have two solutions, one, or none (real), and there are three reliable tools to find them: factoring, the square-root method, and the quadratic formula. The trick is choosing the right tool for the equation in front of you.

In short: set the equation to \(0\), then factor (if it factors), take square roots (if there’s no middle term), or use the quadratic formula \(x = \dfrac{-b \pm \sqrt{b^2 – 4ac}}{2a}\) (always works). For \(x^2 – 5x + 6 = 0\), the roots are \(2\) and \(3\).

The big idea

Three Ways to Find the Roots

Every method finds the same answers; they just suit different equations.

FactoringFast when it factors into nice integers.

Square rootBest when there’s no \(x\) term (like \(x^2 = 9\)).

Quadratic formulaAlways works, even when nothing factors.

Tutor tip: Try factoring first — it’s quickest. If two numbers won’t multiply to \(c\) and add to \(b\), reach for the quadratic formula instead of forcing it.

Worked Examples

The roots are exactly where each parabola crosses the x-axis — shown on every graph below.

Example A — Factoring

Solve \(x^2 – 5x + 6 = 0\).

Find two numbers that multiply to 6 and add to \(-5\): \(-2\) and \(-3\).
Factor: \((x – 2)(x – 3) = 0\).
Set each factor to zero: \(x = 2\) or \(x = 3\).

Answer: \(x = 2\) or \(3\)

Example B — Square-root method

Solve \(x^2 – 9 = 0\).

No middle term — add 9: \(x^2 = 9\).
Take the square root of both sides, keeping \(\pm\).
\(x = \pm 3\).

Answer: \(x = \pm 3\)

Example C — Leading coefficient

Solve \(2x^2 – 7x + 3 = 0\).

Factor: \((2x – 1)(x – 3) = 0\).
Set each factor to zero: \(2x – 1 = 0\) or \(x – 3 = 0\).
Solve: \(x = \tfrac12\) or \(x = 3\).

Answer: \(x = \tfrac12\) or \(3\)

Example D — One repeated root

Solve \(x^2 – 4x + 4 = 0\).

Recognize the perfect square: \((x – 2)^2 = 0\).
Both factors are \(x – 2\), so they give the same value.
One double root: \(x = 2\) — the vertex sits right on the x-axis.

Answer: \(x = 2\)

Where You’ll Use It

Quadratic equations answer “when does it hit zero?” — when a thrown ball lands, when a profit becomes zero (break-even), when an area reaches a target. Setting a quadratic model equal to a value and solving is one of the most common tasks in algebra, physics, and engineering.

Slip-Ups That Cost Easy Points

Not setting it to zero first. Factoring and the formula need \(ax^2 + bx + c = 0\); move everything to one side.
Forgetting the \(\pm\). A square root gives two answers: \(x^2 = 9\) means \(x = 3\) and \(-3\).
Sign errors in the formula. Watch \(-b\) and the \(-4ac\); a single sign flip changes everything.
Reporting only one root. Most quadratics have two solutions — give both unless it’s a perfect square.

Your Turn: Solve

Find all roots, then reveal the answers.

\(x^2 – 7x + 10 = 0\)
\(x^2 – 16 = 0\)
\(x^2 + 5x + 6 = 0\)
\(x^2 – 4x + 4 = 0\)
\(3x^2 – 5x – 2 = 0\)
\(x^2 – 2x – 15 = 0\)

Show answers

\(\color{blue}{x = 2, 5}\)
\(\color{blue}{x = \pm 4}\)
\(\color{blue}{x = -2, -3}\)
\(\color{blue}{x = 2}\)
\(\color{blue}{x = -\tfrac13, 2}\)
\(\color{blue}{x = -3, 5}\)

Keep practicing

Make Your Own Quadratics Worksheet

Generate fresh solve-the-quadratic problems with a full answer key — print or save as a PDF.

New problems every click — never the same sheet twice

Step-by-step answer key so you can self-check

📄 Open the Worksheet Maker ⚡ Quadratic Solver

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Frequently Asked Questions

What are the ways to solve a quadratic equation?

Factoring (fastest when it factors), the square-root method (for equations with no middle term), and the quadratic formula (which always works). All give the same roots.

How many solutions does a quadratic have?

Up to two real solutions. A perfect square gives one (a repeated root), and some quadratics have no real solutions — the discriminant tells you which.

When should I use the quadratic formula?

When the quadratic doesn’t factor with simple integers, or you want a method that always works. Just plug \(a\), \(b\), and \(c\) into \(x = \tfrac{-b \pm \sqrt{b^2 – 4ac}}{2a}\).

Why must I set the equation to zero first?

Factoring relies on the zero-product property (if a product is 0, a factor is 0), and the formula is defined for \(ax^2 + bx + c = 0\). Both need a zero on one side.

Continue Your Study

Ready for the next step? Pick up right where this lesson leaves off:

Up nextGraphing Quadratic FunctionsContinue with Quadratic Functions and EquationsOpen lesson →Up nextSolving a Quadratic Equation by Completing the SquareContinue with Quadratic Functions and EquationsOpen lesson →Up nextSolving a Quadratic Equation by FactoringContinue with Quadratic Functions and EquationsOpen lesson →

Effortless Math Team

6 days ago

Effortless Math Team

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How to Solve a Quadratic Equation? (+FREE Worksheet!)