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How to solve a quadratic equation — step by step (with a worked example)

Q: How do you solve x² − 5x + 6 = 0?

The discriminant is D = (−5)² − 4 · 1 · 6 = 25 − 24 = 1. Then x = (5 ± √1)/2 = (5 ± 1)/2. The solutions are x₁ = 3 and x₂ = 2. Check: 3 + 2 = 5 and 3 · 2 = 6.

Q: What is the quadratic formula?

The general solution x = (−b ± √(b² − 4ac)) / (2a) for any equation of the form ax² + bx + c = 0. It works regardless of the value of a and always gives both roots at once.

Q: What does the discriminant tell you?

The discriminant D = b² − 4ac sits under the square root and controls the number of solutions: D > 0 gives two distinct real solutions, D = 0 gives one repeated solution (the parabola touches the x-axis), and D < 0 gives no real solution.

Q: When does a quadratic equation have no solution?

When the discriminant is negative. For x² + x + 1 = 0, D = 1 − 4 = −3 < 0 — there is no real solution because you cannot take a real square root of a negative number.

Q: When should I factor instead of using the formula?

Factoring is faster when the roots are simple whole numbers. For example x² − 5x + 6 = (x − 2)(x − 3), so the roots are 2 and 3. If you cannot spot a factorization quickly, fall back to the quadratic formula, which always works.

Q: How do I check my answer?

Use Vieta's formulas: the sum of the solutions should equal −b/a and their product should equal c/a. For x₁ = 3 and x₂ = 2: 3 + 2 = 5 = −(−5)/1 and 3 · 2 = 6 = 6/1 — both match.

A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. You solve it with the quadratic formula: x = (−b ± √(b² − 4ac)) / (2a). Worked example: x² − 5x + 6 = 0 → x₁ = 3, x₂ = 2, in four steps. Suitable for Grade 9 / Year 10 algebra.

By Valerij Hofmann · Fullstack Developer

Updated May 29, 2026Intermediate

Quick answer

Put the equation in the form ax² + bx + c = 0 and substitute into the quadratic formula x = (−b ± √(b² − 4ac)) / (2a). The discriminant D = b² − 4ac tells you the number of solutions. For x² − 5x + 6 = 0: D = 1, so x₁ = 3 and x₂ = 2.

At a glance

Summary of this tutorial
Equation	x² − 5x + 6 = 0
Method	Quadratic formula
Steps	4
Answer	x₁ = 3, x₂ = 2
Check (Vieta)	3 + 2 = 5, 3 · 2 = 6 ✓
Grade level	Grade 9 (ages 14–15)

Worked example: x² − 5x + 6 = 0

EXAMPLE

x² − 5x + 6 = 0

Here a = 1, b = −5, c = 6. We first compute the discriminant, then both solutions.

The 4 steps to solve a quadratic equation

These four steps work for any equation of the form ax² + bx + c = 0.

Step 1 · Start
x² − 5x + 6 = 0
Standard form with a = 1, b = −5, c = 6.
Step 2 · Discriminant
D = (−5)² − 4 · 1 · 6 = 1
D = b² − 4ac. Because D > 0, there are two real solutions.
Step 3 · Formula
x = (5 ± √1) / 2
Substitute into the quadratic formula x = (−b ± √D)/(2a).
Step 4 · Answer
x₁ = 3, x₂ = 2
Work out both values: (5 + 1)/2 = 3 and (5 − 1)/2 = 2.
Step 5 · Check
3 + 2 = 5 = −b/a, 3 · 2 = 6 = c/a
Vieta's formulas confirm the solutions.

Why the quadratic formula works

The quadratic formula is simply the result of completing the square, done once in general. You rewrite ax² + bx + c = 0 as a perfect square and take the square root — the ± appears because a square root allows two signs. The expression D = b² − 4ac sits under that root: when it is negative there is no real root and hence no real solution, when it is zero the two solutions merge into one, and when it is positive you get two distinct values.

Practice it yourself

Open the quadratic equation solver

Enter your own a, b, c. See the discriminant and both solutions with every step.

Start practice set

3 questions + boss problem

Frequently asked questions

How do you solve a quadratic equation?

Use four steps: 1) Put it in standard form ax² + bx + c = 0 and read off a, b, c. 2) Compute the discriminant D = b² − 4ac. 3) Substitute into the quadratic formula x = (−b ± √D)/(2a). 4) Work out both solutions and check with Vieta's formulas. For x² − 5x + 6 = 0 this gives x₁ = 3 and x₂ = 2.

How do you solve x² − 5x + 6 = 0?

What is the quadratic formula?

What does the discriminant tell you?

When does a quadratic equation have no solution?

When should I factor instead of using the formula?

How do I check my answer?

Quick answer

At a glance

Worked example: x² − 5x + 6 = 0

The 4 steps to solve a quadratic equation

Step 1 · Start

Step 2 · Discriminant

Step 3 · Formula

Step 4 · Answer

Step 5 · Check

Why the quadratic formula works

Practice it yourself

Frequently asked questions