Quadratic Formula - Explained with Examples
The complete guide to the quadratic formula. Plus: generate it ready for Microsoft Word in one click.
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The quadratic formula - step-by-step solution with worked examples
The quadratic formula solves any equation of the form ax² + bx + c = 0 in one step: x = (−b ± √(b² − 4ac)) / 2a. Instead of guessing factors or completing the square, you substitute the three coefficients - a, b, and c - and compute the two roots directly. It works for every quadratic equation, whether the roots are integers, fractions, irrational numbers, or complex.
The expression under the square root, b² − 4ac, is called the discriminant (often written as Δ). It tells you the nature of the roots before you solve: if Δ > 0, the equation has two distinct real roots; if Δ = 0, there is exactly one repeated root; if Δ < 0, the roots are complex conjugates. Checking the discriminant first saves time and prevents errors when solving by hand.
Worked example: solve 3x² − 5x + 2 = 0. Here a = 3, b = −5, c = 2. The discriminant is (−5)² − 4·3·2 = 25 − 24 = 1. Since Δ > 0, there are two real roots: x = (5 ± 1) / 6, giving x = 1 and x = 2/3. The same approach works for physics (projectile motion, where height = 0 gives a quadratic), engineering (circuit analysis with impedance), and economics (break-even and optimisation models).
Use the generator above to get the quadratic formula - or any variant like the vertex form, the discriminant formula alone, or the factored form - formatted for your Word document, LaTeX file, or presentation.
How it works
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Type a description in plain English - e.g. 'quadratic formula' or 'Maxwell's equations'.
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Frequently asked questions
What is the quadratic formula?
The quadratic formula is x = (−b ± √(b²−4ac)) / 2a. It solves any quadratic equation written as ax² + bx + c = 0 by substituting the coefficients a, b, and c into one standard expression.
How do I use the quadratic formula step by step?
First, rewrite the equation in the form ax² + bx + c = 0. Next, identify a, b, and c. Then calculate the discriminant b²−4ac, substitute everything into the quadratic formula, and simplify the two resulting roots.
What is the quadratic formula example?
For 2x² + 3x − 2 = 0, a = 2, b = 3, and c = −2. The discriminant is 3² − 4·2·(−2) = 25, so x = (−3 ± 5) / 4, which gives x = 1/2 and x = −2.
Can I generate the quadratic formula for Word?
Yes. FormulAI generates the quadratic formula in a format ready for Microsoft Word, so you can paste it as a native equation into assignments, reports, worksheets, or dissertations.
What happens when the discriminant is negative?
If b²−4ac is negative, the square root part becomes imaginary. That means the quadratic equation has two complex roots instead of two real-number solutions.
What is the difference between the quadratic formula and factoring?
Factoring works only when the quadratic expression breaks neatly into simpler binomials. The quadratic formula always works, even when the equation cannot be factorised cleanly.