Quadratic Equation Calculator

Free Online Quadratic Equation Calculator

This quadratic equation calculator solves any quadratic equation instantly using the quadratic formula — type in almost any equation, including ones with brackets, binomial products like (x+3)(x-2) = 0, squared binomials like (x-1)² = 9, or x-terms on both sides, and get the exact roots right away. Along with the answer, you get a fully plotted parabola diagram with the vertex, roots, y-intercept, and axis of symmetry all labeled with their real numeric values, plus a complete step-by-step solution so you can see exactly how the quadratic formula was applied.

Whether you're a student solving algebra homework, a teacher preparing worked examples, or an engineer or analyst who needs a fast, reliable quadratic solver, this tool handles equations with real roots, a single repeated root, or complex (imaginary) roots — no equation is too hard.

Quadratic Equation Formula

A quadratic equation is any equation that can be written in the standard form below, where the highest power of the variable is 2:

• Standard form: ax² + bx + c = 0, where a, b, and c are known numbers and a ≠ 0
• a is the coefficient of x² (it controls how wide or narrow the parabola is and whether it opens upward or downward)
• b is the coefficient of x
• c is the constant term (also the y-intercept of the parabola)
• The quadratic formula: x = (−b ± √(b² − 4ac)) / (2a), which gives the value(s) of x that make the equation true

The Discriminant: b² − 4ac

The expression under the square root in the quadratic formula, b² − 4ac, is called the discriminant, and it tells you exactly what kind of roots the equation has before you even finish solving it:

• If the discriminant is positive (D > 0), the equation has two distinct real roots — the parabola crosses the x-axis at two separate points.
• If the discriminant is exactly zero (D = 0), the equation has exactly one repeated real root — the parabola just touches the x-axis at its vertex.
• If the discriminant is negative (D < 0), the equation has no real roots — instead it has two complex (imaginary) roots, and the parabola never touches the x-axis.

How to Solve a Quadratic Equation Step by Step

Solving any quadratic equation, no matter how it's written, always follows the same sequence of steps:

• Step 1 — Expand any brackets or binomial products, and move every term to one side so the equation equals zero.
• Step 2 — Combine like terms until the equation is in standard form: ax² + bx + c = 0.
• Step 3 — Identify the coefficients a, b, and c directly from the standard form.
• Step 4 — Calculate the discriminant, D = b² − 4ac, to know how many real roots to expect.
• Step 5 — Substitute a, b, and D into the quadratic formula: x = (−b ± √D) / (2a).
• Step 6 — Simplify the + and − cases separately to get the two roots (or the single repeated root if D = 0).
• Step 7 — Check your answer by substituting each root back into the original equation — the left and right sides should be equal.

How to Use This Quadratic Equation Calculator

Using the calculator is simple: type your equation into the input box using x^2 for x², exactly as you would write it on paper — for example, x^2 - 5x + 6 = 0, or something more complex like (x+3)(x-2) = 0 or 3x^2 - 2x - 1 = 5. You don't need to expand brackets or simplify anything first; the calculator automatically expands binomial products, distributes brackets, and moves everything into standard form for you.

As soon as you enter a valid equation, the calculator instantly shows the standard form, the values of a, b, and c, the discriminant, and the final roots — whether real, repeated, or complex. Below that, the parabola diagram plots your exact curve with the vertex, axis of symmetry, y-intercept, and any real roots labeled directly on the graph, so you can see how all the values relate to each other visually. Scroll down further for the complete written step-by-step solution.

Worked Example

Take the equation x² − 5x + 6 = 0. Here a = 1, b = −5, and c = 6. The discriminant is D = (−5)² − 4(1)(6) = 25 − 24 = 1, which is positive, so there are two distinct real roots. Applying the quadratic formula: x = (5 ± √1) / 2 = (5 ± 1) / 2, giving x₁ = 3 and x₂ = 2. You can verify this by substituting back: (3)² − 5(3) + 6 = 9 − 15 + 6 = 0, and (2)² − 5(2) + 6 = 4 − 10 + 6 = 0 — both check out.

A harder example with complex roots: x² + 2x + 5 = 0. Here a = 1, b = 2, c = 5, so D = (2)² − 4(1)(5) = 4 − 20 = −16, which is negative. Since D is negative, the roots are complex: x = (−2 ± √−16) / 2 = (−2 ± 4i) / 2 = −1 ± 2i. So the two roots are x₁ = −1 + 2i and x₂ = −1 − 2i — a complex conjugate pair, and the parabola never crosses the x-axis.

Real Roots, Repeated Roots, and Complex Roots Explained

The nature of a quadratic equation's roots connects directly to the shape and position of its parabola graph. When the discriminant is positive, the parabola dips below (or rises above) the x-axis, crossing it at exactly two points — these crossing points are the two real roots. When the discriminant is zero, the vertex of the parabola sits exactly on the x-axis, so the curve just touches it at one point without crossing — this is the repeated (or double) root.

When the discriminant is negative, the entire parabola stays either completely above or completely below the x-axis, never touching it at all — in this case, the equation still has two solutions, but they are complex numbers involving the imaginary unit i (where i² = −1), not real numbers. Complex roots always come in conjugate pairs, meaning if one root is p + qi, the other is always p − qi.

The Vertex, Axis of Symmetry, and Y-Intercept

Besides the roots, three other features fully describe a parabola. The vertex is the turning point of the curve — its lowest point if the parabola opens upward (when a > 0), or its highest point if it opens downward (when a < 0). The vertex coordinates are given by x = −b / (2a) for the x-coordinate, and by substituting that value back into the equation to find the y-coordinate.

The axis of symmetry is the vertical line that passes through the vertex, at x = −b / (2a) — the parabola is a perfect mirror image on either side of this line. Finally, the y-intercept is simply the point where the curve crosses the y-axis, which happens when x = 0, giving the point (0, c) — the constant term c is always the y-intercept.

Real-World Applications of Quadratic Equations

Quadratic equations aren't just an algebra exercise — they model countless real situations involving curved, accelerating, or area-based relationships:

• Physics and projectile motion: calculating the height of a thrown or launched object over time, since gravity creates a parabolic path.
• Engineering and architecture: designing parabolic arches, satellite dishes, and suspension bridge cables, all of which follow quadratic curves.
• Business and economics: finding the maximum profit or minimum cost point, since revenue and cost functions are often quadratic in terms of price or quantity.
• Area problems: solving for dimensions when you know an area and a relationship between length and width, such as a rectangular garden or room.
• Sports analytics: modeling the trajectory of a ball in basketball, football, or golf to estimate range and peak height.
• Optimization problems: finding the maximum or minimum value of any quantity that depends on a variable squared, from material usage to signal strength.

Quadratic Equations vs. Linear Equations

The key difference between a quadratic and a linear equation is the highest power of the variable. A linear equation only has x to the first power (like 3x + 5 = 11), so its graph is a straight line with at most one solution. A quadratic equation includes an x² term, so its graph is a curved parabola that can have zero real solutions (complex roots), one repeated solution, or two distinct real solutions. If your equation doesn't contain an x² term, this calculator will let you know so you can switch to a linear equation calculator instead.

Tips for Solving Quadratic Equations Faster

A few habits make solving quadratic equations much quicker and reduce mistakes: always move every term to one side first so the equation equals zero, always double-check that you've correctly identified a, b, and c before plugging them into the formula, and always calculate the discriminant first — it immediately tells you whether to expect real or complex roots, which helps you catch calculation errors early. When a quadratic factors nicely (like x² − 5x + 6 = (x − 2)(x − 3)), factoring is often faster than the full formula, but the quadratic formula always works, even when factoring doesn't. And whenever you're unsure about a tricky equation with brackets or terms on both sides, this quadratic equation calculator instantly checks your work and shows every step, so you can compare it against your own working and catch small errors quickly.