Quadratic Formula

Solving ax² + bx + c = 0 for Every Value of x

The quadratic formula solves any equation of the form ax² + bx + c = 0 for x, using only the equation’s three coefficients. Enter a, b, and c, and this calculator finds every value of x that makes the equation true — including the case where the answer involves complex numbers rather than real ones.

A quadratic equation always has exactly two solutions when counted correctly: two distinct real numbers, one real number counted twice (a “repeated root”), or a pair of complex numbers. Which case applies depends entirely on the sign of the discriminant (the part under the square root) — this calculator tells you which one you’ve got, not just the raw numbers.

The Formula

For $\vA{a}x^2 + \vB{b}x + \vC{c} = 0$ (with $\vA{a} \neq 0$):

$$x = \frac{-\vB{b} \pm \sqrt{\vB{b}^2 - 4\vA{a}\vC{c}}}{2\vA{a}}$$

The term under the square root, $\vB{b}^2 - 4\vA{a}\vC{c}$, is called the discriminant:

Worked Example

For $x^2 - 3x + 2 = 0$ ($\vA{a}=1$, $\vB{b}=-3$, $\vC{c}=2$):

1. Discriminant: $(-3)^2 - 4(1)(2) = 9 - 8 = 1$, which is positive, so there are two distinct real roots.
2. $x = \frac{3 \pm \sqrt{1}}{2} = \frac{3 \pm 1}{2}$.
3. $x = \frac{4}{2} = 2$ or $x = \frac{2}{2} = 1$.

Checking: $(1)^2 - 3(1) + 2 = 1 - 3 + 2 = 0$ and $(2)^2 - 3(2) + 2 = 4 - 6 + 2 = 0$ — both solutions check out.

Key Factors to Consider

• A quadratic equation’s solutions are exactly where its graph (a parabola) crosses the x-axis. Two distinct real roots mean the parabola crosses the x-axis at two points; a repeated root means it just touches the axis at one point (its vertex); complex roots mean the parabola never touches the x-axis at all, staying entirely above or below it.
• The discriminant alone tells you the NATURE of the solutions before doing any further calculation. Checking whether $b^2 - 4ac$ is positive, zero, or negative is a quick way to know in advance whether an equation has two real solutions, one repeated solution, or a complex- conjugate pair, without needing to complete the rest of the formula first.
• A quadratic equation’s a coefficient can never be zero, or it stops being quadratic. If a = 0, the $x^2$ term disappears entirely and the equation becomes linear (at most one solution, found by simple algebra) instead of quadratic — this is why the quadratic formula itself is only valid, and only needed, when a is nonzero.
• Complex roots always come in conjugate pairs for a quadratic with real coefficients. When the discriminant is negative, the two solutions always take the form $p + qi$ and $p - qi$ (identical real parts, opposite-signed imaginary parts) — this pairing is a direct consequence of the ± in the formula applied to a negative number under the square root.

Common Mistakes

• Forgetting the ± and reporting only one root. The quadratic formula’s ± sign means there are two separate calculations to do (one with +, one with -) — stopping after the first one misses half the answer whenever the discriminant isn’t exactly zero.
• Treating a negative discriminant as “no solution” instead of a complex solution. A negative discriminant means the equation has no REAL solution, but it still has two valid complex solutions — the equation isn’t unsolvable, just not solvable within the real numbers.
• Entering a = 0 and expecting a quadratic answer. Without a nonzero a, the equation is linear, not quadratic, and the formula’s division by 2a is undefined — a linear equation needs simple algebra instead.

Useful to Know

• Need just a square root or nth root rather than a full equation solve? Square Root / Nth Root Calculator calculates square roots and nth roots directly.
• Working with exponents on their own? Exponent Calculator calculates powers and roots.
• Need a broader calculator for trig, logs, or other functions beyond this equation? Scientific Calculator handles a full range of scientific calculations.