Quadratic Equation Solver
Solve ax² + bx + c = 0 instantly using the quadratic formula.
What this calculator does
Solves any quadratic equation in the form ax² + bx + c = 0, returning both real roots (or complex roots, if applicable) instantly using the quadratic formula.
Who this is for
Students checking homework or studying for an algebra exam, anyone who needs to solve a quadratic as part of a larger problem (physics, engineering, finance), or teachers generating quick answer keys.
Methodology
Solves ax² + bx + c = 0 using the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant, b² − 4ac, determines how many real solutions exist before the square root is even taken.
Worked example
Solving x² − 5x + 6 = 0: here a=1, b=−5, c=6. Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1 (positive, so two real roots). x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2, giving x = 3 and x = 2. You can verify: (x−2)(x−3) = x² − 5x + 6, which matches the original equation.
Interpretation
If the discriminant is positive, the equation has two distinct real roots — the parabola crosses the x-axis at two points. If it's exactly zero, there's one repeated real root — the parabola just touches the x-axis at its vertex. If it's negative, there are no real roots at all — the parabola never touches the x-axis, and the two solutions are complex numbers involving i (the imaginary unit, where i² = −1).
Where Quadratic Equations Show Up in Real Life
Projectile motion: the height of a thrown or launched object over time follows a quadratic equation (h = −½gt² + v₀t + h₀, where g is gravitational acceleration), which is why physics problems about balls, rockets, or thrown objects frequently reduce to solving a quadratic for the time when the object hits the ground (h = 0). Area optimization: if you're maximizing the area of a rectangular garden or enclosure with a fixed amount of fencing, the area as a function of one side length is quadratic, and the maximum occurs at the vertex of that parabola. Profit/revenue modeling: in basic economics, revenue as a function of price (when demand decreases as price increases) often follows a quadratic shape, with the maximum revenue point found the same way — at the vertex, where the derivative (or in this simpler algebraic approach, the axis of symmetry) sits.
Common mistakes
- Forgetting the ± sign. Every quadratic with a positive discriminant has two solutions, not one — both the addition and subtraction case.
- Sign errors on b. If your equation is x² − 5x + 6 = 0, then b = −5, not 5 — a common source of wrong answers.
- Assuming a negative discriminant means "no solution" entirely. It means no real solutions — complex solutions still exist mathematically.
- Forgetting to check for a common factor first. An equation like 2x² − 10x + 12 = 0 solves correctly with the formula directly, but simplifying to x² − 5x + 6 = 0 first (dividing by 2) makes the arithmetic easier and less error-prone.
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Frequently Asked Questions
How do you know how many solutions a quadratic has before solving it?
Check the discriminant (b² − 4ac) first. Positive means two real solutions, zero means exactly one repeated real solution, and negative means no real solutions (only complex ones).
What if the discriminant is negative?
A negative discriminant means the equation has no real roots — only complex roots involving the imaginary unit i. The parabola never crosses the x-axis.
What if a equals zero?
If a is zero, the equation is no longer quadratic — it becomes linear (bx + c = 0) and has at most one solution, not two.
What does the discriminant tell you?
It determines the nature of the roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex roots.
How are quadratic equations used in physics?
Projectile motion (the height of a thrown or launched object over time) follows a quadratic equation, which is why problems about when an object hits the ground typically reduce to solving a quadratic for time.
How do quadratics relate to maximizing area or profit?
When area or revenue is expressed as a function of one variable (like a side length or a price), the relationship is often quadratic, with the maximum value occurring at the vertex of the resulting parabola.