Quadratic Formula Formula

The Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

Quick Example

x^2 - 5x + 6 = 0 \to x = \frac{5 \pm \sqrt{1}}{2} \to x = 3 \text{ or } x = 2

Notation

a, b, c are the coefficients of ax^2 + bx + c = 0. The \pm symbol produces two solutions. \Delta = b^2 - 4ac is the discriminant.

What This Formula Means

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.

Formal View

\forall a,b,c \in \mathbb{R},\; a \neq 0: the roots of ax^2 + bx + c = 0 are x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, lying in \mathbb{R} iff b^2 - 4ac \geq 0, and in \mathbb{C} \setminus \mathbb{R} otherwise.

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.

Common Mistakes

  • Forgetting \pm
  • Errors under the radical
  • Not simplifying fully

Why This Formula Matters

Universal method for solving quadratics when other methods fail.

Frequently Asked Questions

What is the Quadratic Formula formula?

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

How do you use the Quadratic Formula formula?

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

What do the symbols mean in the Quadratic Formula formula?

a, b, c are the coefficients of ax^2 + bx + c = 0. The \pm symbol produces two solutions. \Delta = b^2 - 4ac is the discriminant.

Why is the Quadratic Formula formula important in Math?

Universal method for solving quadratics when other methods fail.

What do students get wrong about Quadratic Formula?

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

What should I learn before the Quadratic Formula formula?

Before studying the Quadratic Formula formula, you should understand: quadratic functions, square roots.

Want the Full Guide?

This formula is covered in depth in our complete guide:

Quadratic Equations: Factoring, Completing the Square, and the Quadratic Formula →
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