Quadratic Equation Solver

Understanding Quadratic Equations

How to Use This Calculator

1. Select your input format from the three tabs at the top (Standard, Vertex, or Factored form)
2. Enter your coefficients in the input fields—a, b, and c for standard form
3. Check the equation preview to make sure you've entered everything correctly
4. Choose a solution method from the dropdown, or select "All Methods" to see every approach
5. Click "Solve Equation" to see your step-by-step solutions, discriminant analysis, and graph

If you know the vertex or roots of your parabola, use those tabs instead—they convert to standard form automatically.

What Is a Quadratic Equation?

A quadratic equation is any equation where x² is the highest power. The name comes from the Latin "quadratus." You can write any quadratic as ax² + bx + c = 0, where a, b, and c are numbers and a isn't zero. If a were zero, you'd have a line—not a parabola.

Graphing a quadratic produces a parabola—a U-shape (or upside-down U). A thrown basketball follows a parabolic arc.

The Quadratic Formula

The quadratic formula lets you solve any quadratic equation. For ax² + bx + c = 0, the solutions are x = (-b ± √(b² - 4ac)) / (2a). It comes from completing the square on the general equation.

That ± symbol means most quadratics have two answers. You get one by adding the square root part, another by subtracting. Negative b up front, the discriminant (b² - 4ac) under the radical, and 2a on the bottom dividing everything.

Understanding the Discriminant

The discriminant is b² - 4ac, the part under the square root in the quadratic formula. It tells you what kind of solutions you'll get:

• Positive discriminant (Δ > 0): Two different real solutions. The parabola crosses the x-axis at two points.
• Zero discriminant (Δ = 0): One repeated real solution. The parabola touches the x-axis at its vertex.
• Negative discriminant (Δ < 0): No real solutions—only complex conjugate pairs involving i.

If the discriminant is a perfect square (1, 4, 9, 16, 25...), your solutions will be rational numbers, and the equation will factor over the integers.

Solving by Factoring

Factoring breaks a quadratic into two linear pieces multiplied together. When a = 1, look for two numbers that multiply to c and add to b. Those become the constants in your factors: (x + m)(x + n) where mn = c and m + n = b. The zero product property applies.

When a ≠ 1, you need two numbers that multiply to ac and add to b, then use grouping. Difference of squares (a² - b²) splits into (a + b)(a - b), and perfect square trinomials (a² + 2ab + b²) equal (a + b)².

Completing the Square

Completing the square turns any quadratic into a perfect square trinomial plus a constant. Isolate the x² and x terms on one side, then add (b/2)² to both sides. This creates a perfect square.

It produces vertex form directly, showing the parabola's turning point. Slower than the quadratic formula for finding roots, but you'll need it to convert between equation forms. Applying this technique to ax² + bx + c = 0 is how you derive the quadratic formula.

Parabolas and Graphs

Every quadratic y = ax² + bx + c graphs as a parabola. Positive a opens upward. Negative a opens downward. Larger |a| means narrower; smaller |a| means wider. Key features:

• Vertex — the turning point
• Axis of symmetry — vertical line through the vertex at x = -b/(2a)
• Y-intercept — at (0, c)
• X-intercepts — the real solutions, if they exist. Roots are equidistant from the axis of symmetry.

Vertex Form

Vertex form is y = a(x - h)² + k, where (h, k) is the vertex. The turning point is in the equation. The h value shifts the parabola horizontally, and k shifts it vertically.

Converting from standard to vertex form requires completing the square. To convert back, expand (x - h)² and distribute a.

Factored Form

Factored form is y = a(x - r₁)(x - r₂), where r₁ and r₂ are the roots. Set each factor to zero to get your solutions. The coefficient a controls direction and width.

Getting to factored form means finding the roots first—by factoring, the quadratic formula, or completing the square. To return to standard form, expand by multiplying.

Real vs Complex Solutions

Real solutions are x-intercepts on the graph—where the parabola crosses or touches the x-axis. Complex solutions involve i (the square root of negative one) and occur when the discriminant is negative. The parabola floats above or below the x-axis without touching it. Complex solutions come in conjugate pairs: if a + bi is a solution, so is a - bi.

Vieta's Formulas

Vieta's formulas link roots directly to coefficients. For ax² + bx + c with roots r₁ and r₂: the sum of roots equals -b/a, and the product equals c/a. These relationships come from expanding (x - r₁)(x - r₂) and comparing to standard form. Use them to check answers or find one root when you know the other.

Special Cases

Perfect square trinomials like x² + 6x + 9 = (x + 3)² give exactly one repeated root. Its discriminant is zero, and the vertex sits on the x-axis. When b = 0, solve x² = -c/a directly; the roots are opposites (or complex conjugates if c/a is positive). When c = 0, factor out x to get x(ax + b) = 0, so one root is zero. Difference of squares: a² - b² factors as (a + b)(a - b).

Common Mistakes to Avoid

Watch for sign errors:

• The negative before b in the quadratic formula. Remember -(-b) = +b.
• Forgetting to add (b/2)² to BOTH sides when completing the square.
• Missing the ± when taking square roots. This loses half your solutions.
• Arithmetic errors in the discriminant. Double-check b² - 4ac.
• Not verifying answers. Substitute solutions back into the original equation.

Applications in Real Life

Projectile motion is a common example. The height equation h(t) = -16t² + v₀t + h₀ (using feet and seconds) models objects thrown vertically under gravity. Set h = 0 to find when it lands; find the vertex to get maximum height.