Step by Step
1
Identify a, b, and c
Match your equation to the standard form ax² + bx + c = 0 and identify each coefficient, being careful with signs.
2
Calculate the discriminant
Compute b² − 4ac first, since this value tells you what kind of solutions to expect before you even finish the formula.
3
Plug into the formula
Substitute a, b, and the discriminant into x = (−b ± √(b² − 4ac)) / 2a.
4
Solve both cases
The ± symbol means you get two answers: one using addition, one using subtraction.
Applied Walkthrough
1
Solve x² + 5x + 6 = 0 using the quadratic formula, where a = 1, b = 5, c = 6.
2
Discriminant: b² − 4ac = 25 − 24 = 1. Since this is positive, expect two distinct real solutions.
3
Plug in: x = (−5 ± √1) / 2 = (−5 ± 1) / 2.
4
Two cases: x = (−5+1)/2 = −2, and x = (−5−1)/2 = −3. Both check out when substituted back into the original equation.
Exam Application
Exams test whether you can correctly identify a, b, c (especially with negative coefficients), compute the discriminant accurately, and correctly produce both solutions from the ± symbol.
⚠ Common Trap
The most common trap is sign errors when identifying b and c from the original equation, especially when the equation has subtraction in it. Always rewrite the equation to clearly match ax² + bx + c = 0 before pulling out your values.
✓ Quick Self-Check
1. What is the quadratic formula?
x = (−b ± √(b² − 4ac)) / 2a.
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2. What does the discriminant (b² − 4ac) tell you?
What kind of solutions to expect — positive means two real solutions, zero means one repeated solution, negative means no real solutions.
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3. Solve x² + 5x + 6 = 0 using the quadratic formula.
x = −2 or x = −3.
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4. Why does the formula have a ± symbol?
Because it produces two possible solutions — one from addition, one from subtraction.
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5. What's a common source of error when identifying a, b, and c?
Sign errors, especially when the original equation includes subtraction.
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