Quadratic Formula: How to Solve Quadratic Equations
The quadratic formula solves any equation of the form ax² + bx + c = 0 (with a ≠ 0), including the ones that won't factor. This guide works through it step by step, shows how the discriminant predicts the kind of roots you'll get, compares it with factoring and completing the square, and lists the errors that account for most wrong answers. A quadratic calculator takes a, b and c and returns the discriminant, both roots and the factored form.
The Formula
x = (−b ± √(b² − 4ac)) ÷ (2a)
Here a is the coefficient of x², b the coefficient of x, and c the constant term. The expression under the square root, b² − 4ac, is the discriminant, and it tells you what kind of solutions to expect:
- Positive: two distinct real roots.
- Zero: one repeated real root.
- Negative: no real roots; the two roots are complex conjugates.
Step by Step: 2x² − 5x − 3 = 0
1. Identify the coefficients. a = 2, b = −5, c = −3. Most errors happen here, usually with a dropped minus sign.
2. Compute the discriminant. b² − 4ac = (−5)² − 4(2)(−3) = 25 + 24 = 49.
3. Substitute. x = (−(−5) ± √49) ÷ (2 × 2) = (5 ± 7) ÷ 4.
4. Evaluate both signs. x₁ = (5 + 7) ÷ 4 = 3 and x₂ = (5 − 7) ÷ 4 = −1/2.
5. Check. 2(3)² − 5(3) − 3 = 18 − 15 − 3 = 0, and 2(−1/2)² − 5(−1/2) − 3 = 1/2 + 5/2 − 3 = 0.
When to Use the Formula vs. Other Methods
Factor first when the numbers are small. x² − 7x + 12 = 0 factors as (x − 3)(x − 4) = 0, so x = 3 or x = 4, faster than any formula.
Complete the square when a = 1 and you want vertex form, which is useful for graphing. For something like 6x² + 11x − 10 = 0, the formula is the more direct route.
Use the formula when roots are irrational or complex. x² + 2x + 5 = 0 has discriminant 4 − 20 = −16, so there are no real roots; the formula gives x = (−2 ± √(−16)) ÷ 2 = −1 ± 2i.
Using the Calculators
The quadratic expression factoring calculator is the most complete tool here: enter a, b and c and it reports the discriminant, both roots (real or complex) and the factored form. For 4x² + 12x + 9 it reports a discriminant of 144 − 144 = 0, the repeated root x = −3/2 and the factored form 4(x + 1.5)².
The polynomial roots calculator, set to degree 2, returns the real roots from a, b and c. The simpler quadratic formula calculator returns only the root with the plus sign, (−b + √D) ÷ 2a, and reports "no real solutions" when D is negative, so use one of the first two when you need both roots.
To check a root by hand, substitute it back into the equation; a root should return 0.
Common Mistakes
- Sign errors with b. In x² − 8x + 7 = 0, b = −8, so −b = +8 in the formula.
- Discriminant arithmetic. For 3x² + 4x + 2 = 0, b² − 4ac = 16 − 24 = −8, not +8. Watch the sign of 4ac when a and c have the same sign.
- Dividing only part of the numerator by 2a. The whole expression −b ± √D is divided by 2a.
- Leaving fractions unsimplified. x = (−6 ± 4) ÷ 6 gives −2/6 and −10/6, which reduce to −1/3 and −5/3.
- Reporting one root. Unless the discriminant is zero, there are two solutions; if you found only one, revisit the ± step.