Quadratic Formula

Algebra
rule

Also known as: solve quadratic, roots formula

Grade 9-12

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A formula giving the exact solutions to any quadratic equation ax^2 + bx + c = 0 directly from its three coefficients. The quadratic formula guarantees you can solve any quadratic equation, even when factoring fails.

This concept is covered in depth in our complete quadratic equations guide, with worked examples, practice problems, and common mistakes.

Definition

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

💡 Intuition

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

🎯 Core Idea

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

Example

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

Formula

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

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.

🌟 Why It Matters

The quadratic formula guarantees you can solve any quadratic equation, even when factoring fails. It is used in physics for projectile motion, in finance for break-even analysis, and in engineering for design optimization.

💭 Hint When Stuck

Write out a, b, and c separately before plugging into the formula, and double-check each sign.

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.

See Also

factoring

🚧 Common Stuck Point

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

⚠️ Common Mistakes

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

Frequently Asked Questions

What is Quadratic Formula in Math?

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

What is the Quadratic Formula formula?

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

When do you use Quadratic Formula?

Write out a, b, and c separately before plugging into the formula, and double-check each sign.

How Quadratic Formula Connects to Other Ideas

To understand quadratic formula, you should first be comfortable with quadratic functions and square roots. Once you have a solid grasp of quadratic formula, you can move on to complex numbers and discriminant.

Want the Full Guide?

This concept is explained step by step in our complete guide:

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

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Visual representation of Quadratic Formula