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.

Tutor-style math help

Solve a Quadratic Equation: what to notice and how to work it

Quadratics skill

Quadratic topics connect an equation, a parabola, roots, and a turning point. Read the form first because each form reveals a different feature.

What to notice first

Standard form helps with formulas, factored form shows roots, and vertex form shows the turning point. A good solution starts by using the form you have.

Common student mistake

Do not assume every quadratic has two real x-intercepts. The discriminant tells whether the real graph crosses the x-axis twice, once, or not at all.

Key formulas and cues

\(ax^2+bx+c=0\)

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

\(x=-\frac{b}{2a}\)

\(y=a(x-h)^2+k\)

A reliable path

Read the formFactored, standard, and vertex forms reveal different features.
Choose the methodFactor when friendly, complete the square for structure, or use the formula when needed.
Connect to the graphRoots are x-intercepts and the vertex is the minimum or maximum point.

Worked examples

Factor and solve

Example: \(x^2-7x+12=0\)

Factor into (x – 3)(x – 4).
Set each factor equal to zero.
Solve both small equations.

Answer: \(x=3\) or \(x=4\)

Find the axis

Example: \(y=2x^2-8x+5\)

Use x = -b/(2a).
Here a = 2 and b = -8.
Compute 8/4.

Answer: \(x=2\)

Open pop-up practice

Try one before moving on

Try: Find the axis of symmetry of \(y=x^2-6x+2\).

Answer: \(x=3\).

Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.

Step-by-Step Solver Worksheet Maker Flashcards

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

Frequently Asked Questions

What does solving a quadratic equation mean?

Solving a quadratic equation means finding the values of x that make ax² + bx + c = 0 true. These values are called roots or solutions, and they are the x-intercepts of the parabola y = ax² + bx + c. A quadratic can have two real roots, one repeated root, or no real roots. For example, x² − 7x + 12 = 0 has roots x = 3 and x = 4, because both make the left side equal zero.

How do you solve quadratic equations step by step?

Start by writing the equation in standard form, ax² + bx + c = 0, with zero on one side. Next, choose a method: factor if the numbers are friendly, take square roots if there is no x term, or use the quadratic formula. Solve, then check each answer in the original equation. For x² + 2x − 15 = 0, factor to (x + 5)(x − 3) = 0, so x = −5 or x = 3.

What is the formula to solve a quadratic equation?

The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a), and it works for any equation of the form ax² + bx + c = 0. The part under the root, b² − 4ac, is the discriminant: positive means two real roots, zero means one, and negative means none. For 2x² + 3x − 2 = 0, the discriminant is 9 + 16 = 25, so x = (−3 ± 5)/4, giving x = ½ or x = −2.

How do you solve a quadratic equation by completing the square?

Move the constant to the right side, make sure the x² coefficient is 1, then add (b/2)² to both sides so the left side becomes a perfect square. Take the square root of both sides and solve. For x² + 6x − 7 = 0, write x² + 6x = 7, add 9 to get (x + 3)² = 16, so x + 3 = ±4. That gives x = 1 or x = −7.

How do you solve quadratic equations with square roots?

The square root method works when the equation has an x² term but no x term. Isolate x², then take the square root of both sides, remembering the ± sign. For 3x² − 48 = 0, add 48 and divide by 3 to get x² = 16, so x = 4 or x = −4. Forgetting the negative root is one of the most common slip-ups with this method.

Is solving quadratic equations hard?

Most students find quadratics manageable once they know the three main methods and when to use each one. The trickiest parts are usually sign errors with negative numbers and forgetting that a quadratic can have two answers. Practice choosing the method first: factoring for friendly numbers, square roots when there is no x term, and the quadratic formula as a backup that always works. Checking answers in the original equation catches most mistakes.

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

2 weeks ago

Effortless Math Team

Related to This Article

Quadratic Equation

What people say about "How to Solve a Quadratic Equation"?

No one replied yet.

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