Quadratic Formula Formula

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

The Formula

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

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

Quick Example

x2βˆ’5x+6=0β†’x=5Β±12β†’x=3Β orΒ x=2x^2 - 5x + 6 = 0 \to x = \frac{5 \pm \sqrt{1}}{2} \to x = 3 \text{ or } x = 2

Notation

aa, bb, cc are the coefficients of ax2+bx+c=0ax^2 + bx + c = 0. The Β±\pm symbol produces two solutions. Ξ”=b2βˆ’4ac\Delta = b^2 - 4ac is the discriminant.

What This Formula Means

A formula giving the exact solutions to any quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 directly from its three coefficients.

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

Formal View

βˆ€a,b,c∈R,β€…β€Šaβ‰ 0\forall a,b,c \in \mathbb{R},\; a \neq 0: the roots of ax2+bx+c=0ax^2 + bx + c = 0 are x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, lying in R\mathbb{R} iff b2βˆ’4acβ‰₯0b^2 - 4ac \geq 0, and in Cβˆ–R\mathbb{C} \setminus \mathbb{R} otherwise.

Worked Examples

Example 1

easy
Solve x2βˆ’5x+6=0x^2 - 5x + 6 = 0 by factoring.

Answer

x=2Β orΒ x=3x = 2 \text{ or } x = 3

First step

1
Find two numbers that multiply to 6 and add to βˆ’5-5: those are βˆ’2-2 and βˆ’3-3.

Full solution

  1. 2
    Factor: (xβˆ’2)(xβˆ’3)=0(x - 2)(x - 3) = 0.
  2. 3
    Set each factor to zero: xβˆ’2=0x - 2 = 0 or xβˆ’3=0x - 3 = 0.
  3. 4
    Solutions: x=2x = 2 or x=3x = 3.
Factoring works when you can find two numbers whose product equals the constant term and whose sum equals the coefficient of xx. The zero product property then gives the solutions.

Example 2

medium
Solve 2x2+3xβˆ’2=02x^2 + 3x - 2 = 0 using the quadratic formula.

Example 3

hard
Solve 2x2βˆ’5xβˆ’3=02x^2 - 5x - 3 = 0 using the quadratic formula.

Common Mistakes

  • Using the formula before setting the equation to zero - rearrange to ax2+bx+c=0ax^2+bx+c=0 first.
  • Mishandling signs of bb in βˆ’b-b - if b=βˆ’4b=-4, then βˆ’b=+4-b=+4.
  • Dropping the Β±\pm and reporting one root - the Β±\pm produces two solutions.

Why This Formula Matters

Factoring only works for tidy quadratics; this formula works for every one, including those with irrational or complex roots. It also hands you the discriminant b2βˆ’4acb^2-4ac, which reveals how many real solutions exist before you finish computing. Recognizing it by "Do I have a quadratic equation set to zero whose exact solutions I need?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from quadratic function and factoring and discriminant in a mixed problem set.

Frequently Asked Questions

What is the Quadratic Formula formula?

A formula giving the exact solutions to any quadratic equation ax2+bx+c=0ax^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?

aa, bb, cc are the coefficients of ax2+bx+c=0ax^2 + bx + c = 0. The Β±\pm symbol produces two solutions. Ξ”=b2βˆ’4ac\Delta = b^2 - 4ac is the discriminant.

Why is the Quadratic Formula formula important in Math?

Factoring only works for tidy quadratics; this formula works for every one, including those with irrational or complex roots. It also hands you the discriminant b2βˆ’4acb^2-4ac, which reveals how many real solutions exist before you finish computing. Recognizing it by "Do I have a quadratic equation set to zero whose exact solutions I need?" β€” rather than by familiar numbers β€” is what lets a student tell it apart from quadratic function and factoring and discriminant in a mixed problem set.

What do students get wrong about Quadratic Formula?

The procedure for quadratic formula is the easy part; the trap is using the formula before setting the equation to zero. Asking "Do I have a quadratic equation set to zero whose exact solutions I need?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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 β†’
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