Quadratic Formula Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Quadratic Formula.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A formula giving the exact solutions to any quadratic equation ax^2 + bx + c = 0 directly from its three coefficients.

When factoring fails, this formula always finds the x-intercepts.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The discriminant (b^2 - 4ac) tells you how many real solutions exist.

Common stuck point: The \pm gives two solutions; when the discriminant b^2 - 4ac < 0, there are no real solutions.

Sense of Study hint: Write out a, b, and c separately before plugging into the formula, and double-check each sign.

Worked Examples

Example 1

easy
Solve x^2 - 5x + 6 = 0 by factoring.

Solution

  1. 1
    Find two numbers that multiply to 6 and add to -5: those are -2 and -3.
  2. 2
    Factor: (x - 2)(x - 3) = 0.
  3. 3
    Set each factor to zero: x - 2 = 0 or x - 3 = 0.
  4. 4
    Solutions: x = 2 or x = 3.

Answer

x = 2 \text{ or } x = 3
Factoring works when you can find two numbers whose product equals the constant term and whose sum equals the coefficient of x. The zero product property then gives the solutions.

Example 2

medium
Solve 2x^2 + 3x - 2 = 0 using the quadratic formula.

Example 3

hard
Solve 2x^2 - 5x - 3 = 0 using the quadratic formula.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Solve x^2 - 9 = 0.

Example 2

hard
Solve x^2 + 4x + 5 = 0 and describe the solutions.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic functionssquare roots