Quadratic Functions

The Quadratic Formula

Solving any quadratic equation using the quadratic formula.

Advanced20 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Solving by Factoring

A Formula That Solves Any Quadratic

x² + 4x + 1 = 0 doesn't factor into nice integer binomials — no pair of integers multiplies to 1 and adds to 4. Before reading on, think about this: since completing the square converted any quadratic into vertex form in an earlier lesson, could that same technique be applied once, in general, to derive a formula that solves every quadratic equation, factorable or not?

That's exactly how the quadratic formula is built: starting from the general equation ax² + bx + c = 0, dividing by a, and completing the square exactly the way vertex form was derived — but keeping a, b, and c as symbols instead of specific numbers. Working through that process (the same steps from the vertex-form lesson, just kept general) eventually isolates x and produces the formula below.

Definition — The Quadratic Formula

For any equation ax² + bx + c = 0 (with a ≠ 0), the solutions are x = (−b ± √(b² − 4ac)) / (2a). The expression under the radical, b² − 4ac, is called the discriminant.

Worked Example — Applying the Quadratic Formula

Solve x² + 4x + 1 = 0. Identify a = 1, b = 4, c = 1. Substitute: x = (−4 ± √(16 − 4))/2 = (−4 ± √12)/2. Simplify √12 = 2√3: x = (−4 ± 2√3)/2 = −2 ± √3. The two solutions are x = −2 + √3 and x = −2 − √3.

Worked Example — Verifying a Solution

Check x = −2 + √3 in x² + 4x + 1 = 0. Substituting is messy by hand, but confirms the pattern: since the formula was derived directly from the original equation through valid algebraic steps, any value it produces is guaranteed to satisfy the equation — the same guarantee that makes solving by factoring trustworthy.

The discriminant, b² − 4ac, predicts how many real solutions an equation has before any further work: a positive discriminant means two distinct real solutions (since ± produces two different values), a discriminant of exactly 0 means one repeated real solution (since ± adds and subtracts nothing), and a negative discriminant means no real solutions at all (since a real number's square root can't be taken).

Worked Example — Predicting the Number of Solutions with the Discriminant

Without fully solving, predict how many real solutions 2x² − 3x + 5 = 0 has. Compute the discriminant: b² − 4ac = (−3)² − 4(2)(5) = 9 − 40 = −31. Since the discriminant is negative, this equation has no real solutions.

x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}

Equation Editor

Edit as raw LaTeX

sum(i, 1, n, i^2) = n(n+1)(2n+1)/6

Constants

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Calculus & discrete math

Greek

\sum_{i=1}^{n} i^{2} = \frac{\operatorname{n}\left(n + 1\right) \cdot \left(2 \cdot n + 1\right)}{6}

Evaluate

Unknown function "n"

Tip

Check the discriminant's sign before fully solving — it instantly tells you whether to expect two solutions, one repeated solution, or no real solutions at all, which is a useful check against a fully worked-out answer.

Common Mistakes

Dividing only the square root term by 2a, instead of the entire numerator, −b ± √(b² − 4ac).
The 2a in the denominator divides the whole numerator — both the −b term and the ± √(...) term — not just the radical part.
Losing track of the sign of b when substituting into the formula, especially when b itself is already negative.
Substitute the value of b carefully, including its own sign, into −b — if b = −3, then −b = −(−3) = 3, a common place for a sign error to slip in.

Key Takeaways

The quadratic formula, derived by completing the square in general, solves any quadratic equation.
The discriminant, b² − 4ac, predicts the number of real solutions before fully solving: positive gives two, zero gives one, negative gives none.
The quadratic formula works even when a quadratic doesn't factor with integer coefficients.

Summary

The quadratic formula guarantees a solution method for every quadratic equation, connecting completing the square, vertex form, and the discriminant into one unified tool. This closes out Algebra I — Geometry continues the mathematical reasoning built here, developing formal proof and spatial relationships in even greater depth.

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