Quadratic Equation Solver

What Is a Quadratic Equation Solver?

The quadratic equation solver finds the roots of any equation in the form ax² + bx + c = 0 using the quadratic formula. This formula, derived by completing the square, has been used for over a thousand years and remains one of the most important tools in algebra. Enter any coefficients and the calculator returns both solutions, whether they are real, repeated, or complex.

The discriminant (b² − 4ac) determines the nature of the roots. When it is positive, the equation has two distinct real roots and the parabola crosses the x-axis at two points. When it equals zero, there is exactly one repeated root and the parabola touches the x-axis at its vertex. When the discriminant is negative, there are no real roots — the parabola does not intersect the x-axis — and the solutions are complex conjugates involving the imaginary unit i.

Beyond finding roots, the calculator provides the vertex of the parabola and its axis of symmetry. These are essential for graphing quadratic functions and for optimisation problems. Engineers use quadratics to model projectile trajectories, economists use them for cost and revenue curves, and physicists apply them in kinematics equations. Understanding the quadratic formula is a gateway to higher mathematics including calculus and differential equations.

How Do You Use This Quadratic Equation Solver?

Enter the coefficients a, b, and c from your equation ax² + bx + c = 0. Click Calculate to see both roots (real or complex), the discriminant value, the vertex, and the axis of symmetry.

1. Identify the coefficients a, b, and c from your equation ax² + bx + c = 0.
2. Enter each coefficient into the corresponding input field.
3. Click Calculate to apply the quadratic formula.
4. Check the discriminant value to understand the nature of the roots.
5. Read both roots from the results — real values or complex numbers.
6. Review the vertex coordinates and axis of symmetry for graphing.

How Does the Quadratic Equation Solver Formula Work?

The formula used: x = (-b ± √(b² - 4ac)) / 2a; Discriminant Δ = b² - 4ac; Vertex = (-b/2a, f(-b/2a))

What Are Some Example Calculations?

For 2x² + 5x − 3 = 0: Δ = 25 + 24 = 49. x = (−5 ± 7) / 4. Roots: x₁ = 0.5, x₂ = −3. Vertex: (−1.25, −6.125).

When Should You Use a Quadratic Equation Solver?

Use the quadratic solver whenever you encounter an equation with an x² term. Common applications include finding the time a projectile hits the ground (physics), calculating break-even points (economics), determining intersection points of curves (geometry), and solving optimisation problems where the cost or profit function is quadratic.

Students at GCSE and A-level use this calculator to verify manual solutions and understand the relationship between the discriminant and the graph. Engineers apply it when designing parabolic arches, satellite dishes, and suspension cables. Programmers use it in game physics for collision detection. Any scenario that produces a second-degree polynomial equation can be solved with this tool.

What Do These Terms Mean?

What Are the Best Tips to Know?

• Always rearrange your equation into standard form ax² + bx + c = 0 before entering coefficients.
• Check the discriminant first — it tells you whether to expect real or complex roots without computing them.
• If a = 0, the equation is linear (bx + c = 0), not quadratic. Solve it as x = −c/b instead.
• Use the vertex form y = a(x − h)² + k for graphing, where (h, k) is the vertex provided by the calculator.
• For integer roots, verify by factoring — if x₁ and x₂ are whole numbers, then ax² + bx + c = a(x − x₁)(x − x₂).

What Mistakes Should You Avoid?

• Entering incorrect signs for b or c — double-check whether the coefficient is positive or negative in the original equation.
• Forgetting to rearrange the equation to equal zero before reading off a, b, and c.
• Assuming the discriminant is always positive — a negative discriminant means complex roots, not an error.
• Dividing by 2a but writing 2 × a incorrectly when a itself is negative — use parentheses to avoid sign errors.