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The Quadratic Formula

For solving ax² + bx + c = 0

ax² + bx + c = 0
x = −b ± b² − 4ac 2a

Gives you the x-values where a parabola crosses the x-axis

What each part means

a The number in front of x² — controls how wide or narrow the parabola is
b The number in front of x — shifts the parabola left or right
c The constant — shifts the parabola up or down (the y-intercept)
± Means you solve twice: once with + and once with −, giving you two solutions

The Discriminant: b² − 4ac

The part under the square root tells you how many solutions to expect:

b² − 4ac > 0 Two real solutions — the parabola crosses the x-axis twice b² − 4ac = 0 One real solution — the parabola just touches the x-axis b² − 4ac < 0 No real solutions — the parabola never reaches the x-axis

How to use it

1
Write your equation in standard form: ax² + bx + c = 0
2
Identify a, b, and c
3
Plug them into the formula
4
Simplify under the square root first (the discriminant)
5
Solve twice: once with + and once with −

Practice

2x² + 5x − 3 = 0
a = 2, b = 5, c = −3
Discriminant: 5² − 4(2)(−3) = 25 + 24 = 49
√49 = 7
x = (−5 + 7) / 4 = 2/4 = 0.5
x = (−5 − 7) / 4 = −12/4 = −3
x = 0.5 or x = −3
x² − 4x + 4 = 0
a = 1, b = −4, c = 4
Discriminant: (−4)² − 4(1)(4) = 16 − 16 = 0
√0 = 0
x = (4 + 0) / 2 = 2
x = 2 (one solution — the parabola just touches the x-axis)
3x² + 2x − 8 = 0
a = 3, b = 2, c = −8
Discriminant: 2² − 4(3)(−8) = 4 + 96 = 100
√100 = 10
x = (−2 + 10) / 6 = 8/6 = 4/3
x = (−2 − 10) / 6 = −12/6 = −2
x = 4/3 or x = −2

Why does this matter?

Quadratics show up any time something curves, accelerates, or follows an arc. Here are real examples:

🏀
Sports & throwing things — When you throw a basketball, its path is a parabola. The quadratic formula tells you exactly when it'll hit the ground or whether it'll clear the hoop. Same for a soccer kick, a baseball, or even a water fountain arc.
🎲
Video games — Every game with jumping, projectiles, or gravity uses quadratic equations behind the scenes. When Mario jumps, the game is solving a parabola to animate his arc. Game developers use this constantly.
💰
Business & money — If you're selling lemonade and you raise the price, you sell fewer cups. There's a sweet spot that maximizes profit — that's the vertex of a parabola. The quadratic formula finds it.
🚗
Stopping distance — When a car brakes, the stopping distance depends on speed squared. Going twice as fast means four times the stopping distance. That's a quadratic relationship, and it's why speed limits exist.
🌍
Satellite dishes & headlights — Parabolic shapes focus signals and light to a single point. Your car headlights, satellite TV dishes, and even telescope mirrors are all designed using quadratic math.
🏭
Architecture — Arches and bridges often use parabolic curves because they distribute weight evenly. The Gateway Arch in St. Louis and many bridge cables follow quadratic shapes.

The big idea: any time you need to find when something hits zero, reaches a maximum, or follows a curved path — that's a quadratic. The formula is the universal key to solving all of them.

Memory trick

x equals negative b
plus or minus the square root
of b squared minus four a c
all over two a

Sing it to the tune of "Pop Goes the Weasel"