The quadratic formula
Example 1: x² − 5x + 6 = 0. D = 25 − 24 = 1, so x = (5 ± 1) ÷ 2, giving x = 3 and x = 2. Factored: (x − 3)(x − 2).
Example 2: 2x² + 4x + 5 = 0. D = 16 − 40 = −24, which is negative, so the roots are complex: x = (−4 ± √−24) ÷ 4 = −1 ± 1.2247i, or exactly (−2 ± √6 i) ÷ 2.
Other ways to solve a quadratic
- Factoring: fastest when the roots are whole numbers or simple fractions. Find two numbers that multiply to ac and add to b.
- Completing the square: rewrite as a(x − h)² + k = 0. It's how the quadratic formula is derived, and it gives the vertex directly.
- Graphing: the real roots are where the parabola crosses the x-axis, as in the graph above.
The quadratic formula always works, which is why it's the safe choice. For the line between two points, use the slope calculator.
Frequently asked questions
What does the discriminant tell you?
b² − 4ac tells you how many real solutions there are: two if it's positive, one if it's zero, and none (two complex solutions) if it's negative. If it's a perfect square, the roots are rational and the quadratic factors nicely.
Can a quadratic have no solution?
It always has two solutions counting complex numbers, but when the discriminant is negative, neither is a real number, so the graph never touches the x-axis.
What if a is 0?
Then it isn't a quadratic. bx + c = 0 is a linear equation with the single solution x = −c ÷ b.
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