How to Solve Quadratic Equations: A Step-by-Step Guide for Students
Quadratic equations look intimidating: squared terms, three coefficients, and a formula that seems to have too many symbols. But here is the secret: every quadratic follows the same pattern, and once you see it, solving them becomes routine.
They also matter beyond the classroom. The curve of a basketball shot, the arc of a fountain, and the path of a satellite all follow quadratic equations. This guide walks you through the two main methods, with every line of working shown.
What Exactly is a Quadratic Equation?
Recognizing the Standard Form
A quadratic equation is any equation that can be written as:
- a is the coefficient of x² (the squared term)
- b is the coefficient of x (the linear term)
- c is the constant, the number with no x
For example, in 2x² − 4x − 6 = 0, we have a = 2, b = −4 and c = −6. Notice that the sign travels with the number.
Why can a never equal 0? If a = 0, the x² term disappears and you are left with bx + c = 0, which is a simple linear equation with only one answer. The x² term is what makes it quadratic and gives it up to two solutions.
Method 1: The Quadratic Formula (The Universal Solvent)
The Formula Breakdown
This formula works on every quadratic equation, even the ones that refuse to factor:
- −b means the opposite sign of b. If b = −4, then −b = 4.
- ± means "plus or minus". You calculate twice, once with + and once with −, to get two answers.
- √(b² − 4ac) is the square root of the discriminant (explained below).
- 2a is the denominator. It divides the entire top line.
What is the Discriminant (b² − 4ac)?
The expression under the square root is called the discriminant, written D. Calculate it first, because it tells you what kind of answers to expect before you finish the problem.
| Value of D | Number of solutions | Example |
|---|---|---|
| D > 0 | 2 distinct real answers | x² − 5x + 6 = 0 → D = 1 |
| D = 0 | 1 real answer (repeated) | x² − 6x + 9 = 0 → D = 0 |
| D < 0 | No real answers (2 imaginary) | x² + x + 1 = 0 → D = −3 |
Method 2: Solving by Factoring (The Speed Strategy)
Finding Factors That Multiply to c and Add to b
When the numbers are friendly, factoring is much faster. You look for two numbers that multiply to c and add to b. Let's solve x² + 5x + 6 = 0.
Here c = 6 and b = 5. List the factor pairs of 6: (1, 6) and (2, 3).
Which pair adds to 5? Only 2 + 3 = 5.
Write the factored form and set each bracket to zero:
x + 2 = 0 or x + 3 = 0
x = −2 or x = −3
Check: (−2)² + 5(−2) + 6 = 4 − 10 + 6 = 0 ✓. If no pair works, don't waste time. Switch to the formula.
Walkthrough Example: Step-by-Step Resolution
Let's solve 2x² − 4x − 6 = 0 using the quadratic formula. Follow along with pen and paper.
Identify a, b and c.
Calculate the discriminant.
D = (−4)² − 4(2)(−6)
D = 16 + 48 = 64
D is positive, so we expect two real answers.
Substitute into the formula.
Solve for both signs.
x = (4 − 8) ÷ 4 = −1
Verify. For x = 3: 2(9) − 12 − 6 = 0 ✓. For x = −1: 2(1) + 4 − 6 = 0 ✓.
Answer: x = 3 or x = −1.
How to Avoid the Most Common Mistakes
- Sign errors with −b. If b = −4, then −b = +4, not −4. A negative of a negative is positive.
- Squaring a negative. (−4)² = 16, not −16. Always put brackets around negative values of b.
- Mishandling −4ac. When c is negative, −4ac becomes positive, as in our example (+48).
- Forgetting the ± symbol. Leaving it out gives you only one of the two answers and costs you marks.
- Dividing only part of the numerator. The 2a divides the whole top line, not just the square root.
- Not setting the equation to zero first. Rearrange to ax² + bx + c = 0 before identifying a, b and c.
Streamline Your Homework Prep
Practicing by hand builds the memory you need in the exam hall, so do the first attempt on paper. Then confirm your working instantly with our Quadratic Equation Calculator. If your answer differs, compare each line to find exactly where the slip happened.
Want more practice? Explore Maths Homework Help for step-by-step solutions, or the Polynomial Calculator for higher-degree problems.