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

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

Worked examples

Factor and solve

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

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

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

Find the axis

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

1. Use x = -b/(2a).
2. Here a = 2 and b = -8.
3. 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.

x

Solve a Quadratic Equation: pop-up practice

Answer these quick questions, then use the feedback to decide which part of the lesson to review.

Choose an answer to begin.

1. The graph of a quadratic is a:

AParabolaBLineCCircle

2. For \(x^2-9=0\), the solutions are:

A\(x=\pm3\)B\(x=9\)C\(x=-9\)

3. The vertex tells you the:

AMinimum or maximumBOnly the y-interceptCSlope

Step-by-Step SolverWorksheet MakerFlashcards

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 \(3\).

Worked Examples

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

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

Your Turn: Solve

Find all roots, then reveal the answers.

Frequently Asked Questions

Related Topics

Continue Your Study

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