The Quadratic Formula: How to Use It

The quadratic formula solves any equation of the form ax² + bx + c = 0. You don't need to factor — just plug in a, b, and c.

The ± means you get two answers: one with + and one with −. Most quadratics have 0, 1, or 2 real solutions.

Step-by-step: 2x² + 5x − 3 = 0

Step 1: Identify a, b, and c.

ax² + bx + c = 0 → a = 2, b = 5, c = −3

Step 2: Plug into the formula.

Step 3: Split into two answers.

Solutions: x = ½ or x = −3

Check: 2(½)² + 5(½) − 3 = ½ + 2.5 − 3 = 0 ✓

The Discriminant: b² − 4ac

The expression under the square root tells you how many real solutions exist:

• b² − 4ac > 0 → two real solutions
• b² − 4ac = 0 → one real solution (a "double root")
• b² − 4ac < 0 → no real solutions (solutions are complex numbers)

Example: x² − 4x + 4 = 0

a = 1, b = −4, c = 4. Discriminant = (−4)² − 4(1)(4) = 16 − 16 = 0. One solution.

So x² − 4x + 4 = (x − 2)², and the only solution is x = 2.

When to Use the Formula

Use the quadratic formula when:

• Factoring is difficult or not obvious
• The equation has decimals or messy numbers
• You want a reliable method that always works

For simple quadratics, factoring can be faster.

Common mistakes

Sign errors with −b: If b = −5, then −b = −(−5) = +5. Watch the double negative.

Forgetting the ±: The formula gives two solutions (or one, or none). Don't stop after just the + case.

Wrong discriminant: b² − 4ac, not b² − 4a. Make sure to multiply all three: 4 × a × c.

The discriminant

The expression under the square root, b² − 4ac, is called the discriminant. It tells you how many solutions exist:

• Positive: two real solutions
• Zero: exactly one real solution (a repeated root)
• Negative: no real solutions (complex roots)

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