The Quadratic Formula

Learn to solve any quadratic equation using the quadratic formula.

Intermediate25 minLesson

Definition

The quadratic formula solves any equation of the form

ax^2 + bx + c = 0

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

Where:

$a$ , $b$ , and $c$ are the coefficients from the equation
$a \neq 0$ (otherwise it's not quadratic)
The $\pm$ symbol means there are usually two solutions

The expression under the square root,

b^2 - 4ac

, is called the discriminant.

Try it now

In the equation

x^2 + 5x + 6 = 0

, what are the values of

a

b

, and

c

Worked Examples

Solve $x^2 + 6x + 5 = 0$ using the quadratic formula.

Identify a, b, and c

Comparing to $ax^2 + bx + c = 0$ : $a = 1$ , $b = 6$ , $c = 5$

Calculate the discriminant

$b^2 - 4ac = 6^2 - 4(1)(5) = 36 - 20 = 16$ → $D = 16$

Apply the formula

$x = \frac{-6 \pm \sqrt{16}}{2(1)} = \frac{-6 \pm 4}{2}$ → $x = \frac{-6 \pm 4}{2}$

Find both solutions

$x_1 = \frac{-6 + 4}{2} = \frac{-2}{2} = -1$ $x_2 = \frac{-6 - 4}{2} = \frac{-10}{2} = -5$ → $x = -1$ or $x = -5$

Answer: $x = -1$ or $x = -5$

Common Mistakes

Forgetting to make b negative when it's already negative

Why it's wrong: If $b = -7$ , then $-b = -(-7) = 7$ , not $-7$ . The formula says $-b$ , so you negate whatever b is.

Correct: Always write $-b$ first, then substitute: if $b = -7$ , write $-(-7) = 7$

Calculating $b^2$ incorrectly when b is negative

Why it's wrong: Students sometimes write $-7^2 = -49$ instead of $(-7)^2 = 49$

Correct: The entire coefficient is squared: $(-7)^2 = (-7) \times (-7) = 49$

Dividing only part of the numerator by 2a

Why it's wrong: Writing $\frac{-b}{2a} \pm \sqrt{b^2-4ac}$ instead of $\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

Correct: The entire numerator $(-b \pm \sqrt{...})$ is divided by $2a$

Stopping at one solution

Why it's wrong: The $\pm$ means plus OR minus, giving two potential solutions

Correct: Always calculate both: $x_1 = \frac{-b + \sqrt{D}}{2a}$ and $x_2 = \frac{-b - \sqrt{D}}{2a}$

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Practice Problems

15 problems

Problem 1 of 15

Easy

In the equation $x^2 + 5x + 6 = 0$ , what are the values of $a$ , $b$ , and $c$ ?

Why It Matters

The quadratic formula is one of the most powerful tools in algebra because:

Universal: It works for ANY quadratic equation, even when factoring doesn't
Predictable: Follow the same steps every time
Real-world applications: Physics (projectile motion), engineering (parabolic structures), finance (profit optimization)

While factoring is faster when it works, many real-world problems have "messy" numbers that don't factor nicely. The quadratic formula handles them all!

Real World Applications

Projectile Motion

When you throw a ball, its height follows a quadratic equation. The formula helps find when it lands.

Example:

A ball thrown upward has height $h = -5t^2 + 20t + 1$ meters after $t$ seconds. To find when it hits the ground ( $h = 0$ ), solve $-5t^2 + 20t + 1 = 0$ using the formula.

1Try It Yourself

A rocket's height is modeled by $h = -4t^2 + 32t$ meters.

When does the rocket return to the ground?

Step 1: Write the mathematical expression

Set $h = 0$ and identify the coefficients:

Business Profit Optimization

Companies use quadratic equations to model profit and find break-even points.

Example:

A company's profit is $P = -2x^2 + 100x - 800$ dollars, where $x$ is units sold. To find break-even points (where $P = 0$ ), use the quadratic formula.

2Try It Yourself

A shop's weekly profit is $P = -x^2 + 50x - 400$ dollars for selling $x$ items.

At what sales levels does the shop break even?

Step 1: Write the mathematical expression

Solve $-x^2 + 50x - 400 = 0$ :

Key Takeaways

1The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
2It works for any equation in the form $ax^2 + bx + c = 0$
3The discriminant $D = b^2 - 4ac$ tells you how many solutions exist
4If $D > 0$ : two distinct real solutions; if $D = 0$ : one repeated solution; if $D < 0$ : no real solutions
5Always check your answers by substituting back into the original equation

Frequently Asked Questions

Use the quadratic formula when: (1) the equation doesn't factor easily, (2) the coefficients are large or decimals, or (3) you want a reliable method that always works. Factoring is faster when it works, but the formula is more universal.

Glossary

Quadratic equation: An equation of the form $ax^2 + bx + c = 0$ where $a \neq 0$
Discriminant: The expression $b^2 - 4ac$ that determines the number and type of solutions
Coefficient: The numerical factor of a term (e.g., in $3x^2$ , the coefficient is 3)
Root/Solution: A value of $x$ that makes the equation true (where the parabola crosses the x-axis)

Formula Card

Quadratic Formula

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

Solves any equation $ax^2 + bx + c = 0$

Discriminant

D = b^2 - 4ac

Determines number and type of solutions

Two solutions

D > 0

Discriminant positive: two distinct real roots

One solution

D = 0

Discriminant zero: one repeated root

No real solutions

D < 0

Discriminant negative: no real roots