Tutorial
How to Solve Quadratic Equations: 3 Methods with Examples
Published 14 March 2026 · 10 min read
A quadratic equation is any equation that can be written in the standard form , where , , and are constants and . These equations appear throughout mathematics, physics, engineering, and economics. Whether you are modelling projectile motion, optimising profit, or solving geometry problems, quadratic equations are essential.
There are three main methods for solving quadratics: the quadratic formula, factoring, and completing the square. Each method has its strengths. In this guide, you will learn all three approaches with fully worked examples so you can choose the right tool for any problem.
Key Concepts
Standard Form
Every quadratic equation can be rearranged into standard form:
The coefficient is the quadratic coefficient, is the linear coefficient, and is the constant term.
The Discriminant
The discriminant tells you how many real solutions exist:
- : two distinct real solutions
- : one repeated real solution
- : no real solutions (two complex solutions)
Method 1: The Quadratic Formula
The quadratic formula works for every quadratic equation. It is derived by completing the square on the general form and gives:
Example 1: Solve
Identify: , , .
So or .
Example 2: Solve
Here , , .
The discriminant is zero, so there is one repeated solution: .
Method 2: Factoring
Factoring is the quickest method when it works. The idea is to rewrite the quadratic as a product of two binomials. If factors as , then by the zero product property, either or .
Example 3: Solve
Find two numbers that multiply to 6 and add to 5. Those numbers are 2 and 3.
So or .
Example 4: Solve
Factor out the greatest common factor first:
So or .
Method 3: Completing the Square
Completing the square transforms the equation into the form , which you can solve by taking square roots. This method always works and is particularly useful for deriving the vertex form of a parabola.
Step-by-step process
- Move the constant term to the right side of the equation.
- If , divide every term by .
- Take half the coefficient of , square it, and add it to both sides.
- Factor the left side as a perfect square trinomial.
- Take the square root of both sides and solve for .
Example 5: Solve
Move the constant:
Half of 6 is 3, and . Add 9 to both sides:
Take the square root:
So or .
Example 6: Solve
Divide by 2:
Move the constant:
Half of is , and . Add 4 to both sides:
So or .
Choosing the Right Method
| Method | Best when | Limitation |
|---|---|---|
| Quadratic formula | Always works, messy coefficients | More computation |
| Factoring | Small integer coefficients | Not always factorable over integers |
| Completing the square | Vertex form needed, deriving formulas | More steps than factoring |
Common Mistakes
Forgetting the negative root
The in the quadratic formula means there are usually two solutions. Always compute both.
Sign errors in the discriminant
Be careful with when or is negative. A negative times a negative is positive.
Not rearranging to standard form
Before applying any method, make sure the equation equals zero. Move all terms to one side first.
Dividing by and losing a solution
If you have , do not divide both sides by . You will lose the solution . Factor instead.
Try It Yourself
Ready to practise? Use our free quadratic formula calculator to check your work instantly. Enter your coefficients and see the full step-by-step solution.
Open Quadratic Formula CalculatorRelated Articles
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