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How to Solve Quadratic Equations
Four methods that always work, when to use each one, and how to check your answer.
A quadratic equation is any equation that can be written as ax² + bx + c = 0, where a is not zero. Its graph is a parabola, and the solutions (also called roots or zeros) are the x-values where that parabola crosses the x-axis. A quadratic has at most two real solutions, and there are four standard ways to find them.
Method 1: Factoring
Factoring is the fastest method when the numbers are friendly. Rewrite the left side as a product of two brackets, then use the zero product rule: if two things multiply to zero, at least one of them must be zero.
Example: Solve x² − 5x + 6 = 0.
Look for two numbers that multiply to 6 and add to −5. Those are −2 and −3, so the equation becomes (x − 2)(x − 3) = 0.
Either x − 2 = 0 or x − 3 = 0, so x = 2 or x = 3.
When a is not 1, factoring takes one more step. For 2x² + 3x − 2 = 0, the factors are (2x − 1)(x + 2) = 0, which gives x = 1/2 or x = −2. If you cannot find the factors within a minute, switch to the quadratic formula.
Method 2: The quadratic formula
The quadratic formula solves every quadratic equation, including ones that do not factor nicely:
Example: Solve x² + 4x − 1 = 0.
Here a = 1, b = 4 and c = −1. Then b² − 4ac = 16 + 4 = 20, and √20 = 2√5.
x = (−4 ± 2√5) / 2, so x = −2 + √5 ≈ 0.236 or x = −2 − √5 ≈ −4.236.
Try it in CasCalc →A common mistake is forgetting that the whole top, not just the square root, is divided by 2a. Another is losing the sign of c: when c is negative, −4ac becomes positive.
Method 3: Completing the square
Completing the square rewrites the equation so that one side is a perfect square. It is also how the quadratic formula is derived, and it is the method to use when you need the vertex of the parabola.
Example: Solve x² + 6x + 5 = 0.
Move the constant: x² + 6x = −5. Take half of 6, which is 3, and square it to get 9. Add 9 to both sides: x² + 6x + 9 = 4.
The left side is now (x + 3)², so (x + 3)² = 4 and x + 3 = ±2. That gives x = −1 or x = −5.
Method 4: Graphing
Graph y = ax² + bx + c and read off where the curve meets the x-axis. Graphing is a great way to check an answer or to see how many solutions to expect before doing any algebra. In CasCalc, typing an equation in x both solves it and marks the roots on the graph, so you get the exact answer and the picture together.
How many solutions? The discriminant
The expression under the square root, D = b² − 4ac, is called the discriminant. It tells you what kind of answers to expect before you solve:
| Discriminant | Solutions | Graph |
|---|---|---|
| D > 0 | Two different real solutions | Crosses the x-axis twice |
| D = 0 | One repeated real solution | Touches the x-axis once |
| D < 0 | No real solutions (two complex ones) | Never meets the x-axis |
Example: For x² + 2x + 5 = 0, D = 4 − 20 = −16, which is negative. There are no real solutions. Using the formula with complex numbers gives x = −1 ± 2i.
Which method should you use?
- Factoring when the coefficients are small whole numbers and the factors are easy to spot.
- The quadratic formula for everything else. It never fails.
- Completing the square when you also need the vertex, or when your teacher asks for it.
- Graphing to check your work and to estimate answers.
Always check your answer
Substitute each solution back into the original equation. For x = 3 in the first example: 3² − 5·3 + 6 = 9 − 15 + 6 = 0. It works. Checking takes ten seconds and catches most sign errors.
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