Quadratic Equation Solver | Free ax² + bx + c = 0 Tool
Solve any quadratic equation ax² + bx + c = 0. Find the discriminant, real or complex roots, vertex coordinates, and see the complete step-by-step solution.
What Is a Quadratic Equation?
A quadratic equation has the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. It's called "quadratic" from the Latin "quadratus" (square) because the variable appears squared.
Quadratic equations appear everywhere in math, science, and engineering. Any time you see a squared variable — whether it's projectile motion, profit optimization, or circuit design — you're dealing with a quadratic.
The Quadratic Formula
The quadratic formula works for every quadratic equation. Memorize it — it always gives you the answer.
Example: x² − 5x + 6 = 0
a = 1, b = −5, c = 6
Discriminant: 25 − 24 = 1
x = [5 ± 1] / 2 → x = 3 or x = 2
Three Methods to Solve Quadratic Equations
There are three main ways to solve a quadratic equation. Each has its strengths depending on the equation.
Method 1: The Quadratic Formula
x = [−b ± √(b² − 4ac)] / 2a. Works for every quadratic equation. Example: x² − 5x + 6 = 0. a=1, b=−5, c=6. Discriminant: 25 − 24 = 1. x = [5 ± 1] / 2 → x = 3 or x = 2.
Method 2: Factoring
Find two numbers that multiply to c and add to b. x² − 5x + 6: numbers are −2 and −3. (x − 2)(x − 3) = 0 → x = 2 or x = 3. Best for simple equations with integer roots.
Method 3: Completing the Square
Transform ax² + bx + c = 0 into (x + p)² = q. Useful when factoring isn't obvious and you want to understand the graph's vertex form.
The Discriminant (Δ = b² − 4ac)
The discriminant tells you the number and type of solutions without solving the equation. It's the expression under the square root in the quadratic formula.
Δ > 0
Two distinct real roots — the parabola crosses the x-axis twice
Δ = 0
One repeated root — the parabola touches the x-axis once
Δ < 0
Two complex roots — the parabola never crosses the x-axis
Real-World Applications of Quadratic Equations
Quadratic equations aren't just abstract math — they model real physical and business phenomena.
Physics: Projectile Motion
Quadratic equations describe the path of any object under gravity. The height equation h = −4.9t² + v₀t gives a parabola — solve for t when h = 0 to find when the object lands.
Business: Profit Maximization
Revenue and cost are often quadratic functions. The vertex of the profit parabola gives the price or quantity that maximizes profit.
Engineering: Circuit Design
Circuit analysis uses quadratic equations to calculate component values. RLC circuits, signal filtering, and resonance frequency all rely on quadratic formulas.
Real-World Quadratic Examples
Example 1: Throwing a Ball
A ball is thrown upward at 20 m/s from a 45m cliff. Height: h(t) = −4.9t² + 20t + 45. When does it hit the ground? Solve −4.9t² + 20t + 45 = 0 using the quadratic formula.
t = [−20 ± √(400 + 882)] / −9.8 = [−20 ± 35.8] / −9.8
t ≈ 5.7 seconds (the negative root is discarded as it represents time before the throw).
Example 2: Maximizing Profit
A company's profit is P(x) = −2x² + 120x − 800, where x is units sold (in hundreds). What quantity maximizes profit?
Vertex x = −b/(2a) = −120/(2×−2) = 30
Profit is maximized at 3,000 units. P(30) = −2(900) + 3600 − 800 = $1,000 (hundred).
Example 3: Fencing a Garden
You have 60m of fencing and want to enclose a rectangular garden against a wall (3 sides only). Area = x(60 − 2x) = −2x² + 60x. What dimensions maximize area?
Vertex x = −60/(2×−2) = 15m. Width = 15m, Length = 30m.
Maximum area = 450 m².
Quadratic Equation Benchmarks
Common quadratic patterns and their properties. Recognize these to solve faster.
| Pattern | Example | Roots |
|---|---|---|
| Perfect square | x² − 6x + 9 = 0 | x = 3 (double root) |
| Difference of squares | x² − 9 = 0 | x = 3, x = −3 |
| Simple factoring | x² − 5x + 6 = 0 | x = 2, x = 3 |
| No real roots | x² + x + 1 = 0 | Complex: (−1 ± i√3)/2 |
| Zero c term | x² − 4x = 0 | x = 0, x = 4 |
| Zero b term | x² − 16 = 0 | x = 4, x = −4 |
Common Quadratic Equation Mistakes
Mistake 1: Forgetting the ± in the Formula
The quadratic formula gives two solutions — one with + and one with −. Forgetting the ± means you only find one root. For x² − 5x + 6 = 0, the solutions are x = 3 AND x = 2, not just one.
Mistake 2: Sign Errors with Negative b
When b is negative (e.g., x² − 5x + 6 = 0, b = −5), the formula becomes x = [5 ± √(25 − 24)] / 2, not x = [−5 ± ...]. The double negative cancels out. Always identify a, b, c with their signs first.
Mistake 3: Not Checking Your Solutions
Always substitute each solution back into the original equation. If x = 3 in x² − 5x + 6 = 0: 9 − 15 + 6 = 0 ✓. If you get a non-zero result, you made an arithmetic error.
Mistake 4: Forgetting the Discriminant Under the Square Root
The discriminant is b² − 4ac, not b² + 4ac. The minus sign matters. If you use plus, you'll get the wrong number of roots and incorrect solutions.
Key Takeaways
- Quadratic formula works for every quadratic equation — memorize it
- Factoring is faster for simple equations — look for integer pairs that multiply to c and add to b
- The discriminant tells you how many real solutions exist
- Always check your solutions by substituting back into the original equation
Frequently Asked Questions
What is the quadratic formula?
x = (-b ± √(b² - 4ac)) / 2a. It solves any equation of the form ax² + bx + c = 0.
How do you know how many solutions a quadratic has?
Calculate the discriminant (b² - 4ac). Positive = 2 real roots, zero = 1 repeated root, negative = 2 complex roots.
What is the vertex of a parabola?
The vertex is the turning point. x = -b/(2a), y = f(x). It's the maximum if a < 0, minimum if a > 0.
Can a quadratic equation have no real solutions?
Yes, when the discriminant is negative. The solutions are complex numbers with imaginary parts.
When should I use factoring vs the quadratic formula?
Use factoring when the equation is simple and factors are obvious (small integers). Use the quadratic formula for any equation — it always works. Factoring is faster for simple cases; the formula is reliable for everything.
What are complex solutions?
Complex solutions occur when the discriminant is negative. The square root of a negative number is imaginary (i), giving solutions in the form a + bi. These don't appear on a standard graph but are valid mathematical solutions.
How do I check my quadratic solutions?
Substitute each solution back into the original equation. Both should make the left side equal zero. If you used the quadratic formula correctly, both solutions will satisfy the equation.
Quadratic Equation Solver Pros & Cons
Pros
- ✅ Real & complex root support
- ✅ Discriminant & vertex display
- ✅ Step-by-step solution shown
- ✅ Free forever — no subscriptions
- ✅ Works offline after page load
Cons
- ✗ Quadratic equations only (ax² + bx + c)
- ✗ No graphical parabola plot
Reviewed by Shahid
Content Reviewer & Calculator SpecialistContent reviewer specializing in marketing, finance, health, and math calculators on GM Calculator.