Quadratic Formula

📖 What is the Quadratic Formula?

The quadratic formula is used to find the roots (solutions) of any quadratic equation of the form:

ax² + bx + c = 0

where a ≠ 0. The formula always works — even when the equation can't be factored easily.

🔤 What Each Variable Means

x

The unknown (solution) What we're solving for — the value(s) that make the equation true

a

Coefficient of x² The number multiplied by x² (cannot be zero)

b

Coefficient of x The number multiplied by x (the middle term)

c

Constant term The number with no variable (the last term)

±

Plus or Minus Gives two solutions: one with + and one with −

📝 Step-by-Step Example

Solve: x² - 5x + 6 = 0

1

Identify a, b, and c

Look at the equation: x² - 5x + 6 = 0

a = 1, b = -5, c = 6

2

Calculate the discriminant (b² - 4ac)

This tells us how many solutions exist

b² - 4ac = (-5)² - 4(1)(6) = 25 - 24 = 1

3

Plug into the formula

x = (-b ± √(b²-4ac)) / 2a

x = (5 ± √1) / 2 = (5 ± 1) / 2

4

Calculate both solutions

Use + first, then −

x₁ = (5 + 1) / 2 = 3 x₂ = (5 - 1) / 2 = 2

✓

Answer: x = 3 or x = 2

Verify: (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓

🧮 Interactive Calculator

Enter your values for ax² + bx + c = 0

a (x² coefficient)

b (x coefficient)

c (constant)

Result will appear here

💡 The Discriminant (b² - 4ac)

2️⃣

b² - 4ac > 0
Two different real solutions

1️⃣

b² - 4ac = 0
One repeated real solution

❌

b² - 4ac < 0
No real solutions (complex numbers)

📖 What is the Quadratic Formula?

🔤 What Each Variable Means

📝 Step-by-Step Example

🧮 Interactive Calculator

💡 The Discriminant (b² - 4ac)

🔗 Related Formulas

🎯 Quick Fact