Quadratic Formula Guide: Solving ax² + bx + c = 0一元二次方程求根公式指南：解 ax² + bx + c = 0
Every quadratic equation — every parabola, every projectile trajectory, every optimization with a squared term — collapses to the same five-symbol formula. The quadratic formula is the most-used closed-form solution in algebra after the four basic operations. It always works, even when factoring is awkward or impossible. This guide explains the formula, walks through the algebra, and shows how to read the result.
What Is a Quadratic Equation?
A quadratic equation is any equation of the form:
- a — coefficient of x² (must not be 0, otherwise it's not quadratic)
- b — coefficient of x
- c — constant term
- x — the unknown you're solving for
The graph is always a parabola. The equation has at most two real solutions (called "roots").
Some real-world examples: a ball's height over time, the break-even point in business, the dimensions of a rectangle given area and perimeter. Whenever a variable is squared, you have a quadratic.
The Quadratic Formula
For any quadratic equation, the solutions are:
Plug in a, b, c; the ± gives both roots. Works for every quadratic — even when factoring fails.
The formula is universal: every quadratic equation that has solutions can be solved by this single expression. The two solutions come from the ± symbol — one root from adding the square root, one from subtracting.
The Discriminant — Your Crystal Ball
The piece under the square root, b² − 4ac, is called the discriminant. It tells you everything about the nature of the solutions before you compute them:
- b² − 4ac > 0: Two distinct real roots. The parabola crosses the x-axis twice.
- b² − 4ac = 0: One repeated real root (a "double root"). The parabola just touches the x-axis.
- b² − 4ac < 0: No real roots; two complex roots. The parabola doesn't cross the x-axis at all.
For the equation 2x² + 5x − 3 = 0: discriminant = 25 − 4(2)(−3) = 25 + 24 = 49, positive — expect two real roots.
A Worked Example
Solve: 2x² + 5x − 3 = 0
Here, a = 2, b = 5, c = −3. Plug in:
x = [ −5 ± √(25 − 4(2)(−3)) ] / 2(2)
= [ −5 ± √(25 + 24) ] / 4
= [ −5 ± √49 ] / 4
= [ −5 ± 7 ] / 4x₁ = (−5 + 7) / 4 = 2/4 = 0.5
x₂ = (−5 − 7) / 4 = −12/4 = −3
Verify by substitution: 2(0.5)² + 5(0.5) − 3 = 0.5 + 2.5 − 3 = 0 ✓; 2(−3)² + 5(−3) − 3 = 18 − 15 − 3 = 0 ✓.
When to Factor Instead
The quadratic formula always works, but factoring is faster when it works. Try factoring first when the coefficients are small and the discriminant is a perfect square (like 49 in the example above). The roots 0.5 and −3 correspond to the factorization:
Setting each factor to zero gives the same roots: 2x − 1 = 0 → x = 0.5; x + 3 = 0 → x = −3.
When the discriminant isn't a perfect square (like 17 or 73), factoring produces ugly fractions — that's when the formula saves you.
Real-World Uses
Three frequent applications:
| Scenario | Quadratic setup |
|---|---|
| Projectile motion | Height h(t) = h₀ + v₀t − ½gt². Set h(t) = 0 to find when it lands. |
| Profit maximization | Profit = (price − cost) × quantity. With linear demand, quantity depends on price, giving a squared term. |
| Area problems | Given perimeter, find the rectangle dimensions that maximize area (or any area/perimeter combination). |
Worked example: a ball is thrown upward at 30 m/s from a 40 m cliff. When does it hit the ground? h(t) = 40 + 30t − 4.9t². Set to zero:
4.9t² − 30t − 40 = 0
t = [30 ± √(900 + 784)] / 9.8 = [30 ± √1684] / 9.8 ≈ [30 ± 41.04] / 9.8
t = 7.25 seconds (taking the positive root)
When There Are No Real Solutions
If the discriminant is negative — say x² + 4x + 7 = 0 — the formula still works, but you get complex numbers:
The two complex roots are conjugates: −2 + i√3 and −2 − i√3. Always comes in pairs when coefficients are real.
Geometrically, the parabola y = x² + 4x + 7 never touches the x-axis (its vertex is at y = 3, above the axis). The complex roots represent the algebraic solutions that don't correspond to any real x.
For ax² + bx + c = 0 with roots r₁ and r₂, the sum r₁ + r₂ = −b/a and the product r₁ × r₂ = c/a. Use these to check your work: for 2x² + 5x − 3 = 0 with roots 0.5 and −3, the sum is −2.5 = −5/2 ✓ and the product is −1.5 = −3/2 ✓.
Try It
Use the Quadratic Formula Calculator to:
- Solve any ax² + bx + c = 0, with real or complex roots.
- See the discriminant value and what it predicts before you compute.
- Verify with Vieta's formulas — sum and product of the roots.
The calculator accepts a, b, c in any format, displays both roots (real or complex), and shows the discriminant, vertex, and factored form where possible.