🧮 Quadratic Equation Solver

Solve quadratic equations of the form ax² + bx + c = 0. Enter the coefficients a, b, and c to see the discriminant, the type of roots (real or complex), and the exact solutions with step-by-step formulas.

Quadratic equation solver results

Enter values for a, b, and c, then choose Solve Quadratic Equation to see the discriminant, root type, and exact solutions. Remember that a must be non-zero for the equation to be quadratic.

Quadratic solution breakdown for your equation

After you solve an equation, this section will show the coefficients, discriminant, root type, and solutions in one place so you can copy them into homework, reports, or notes.

Quadratic equation inputs and key terms

This calculator works with equations of the form ax² + bx + c = 0. Here's what each piece means.

• Coefficient a: The number multiplying x². It controls how “wide” or “narrow” the parabola is, and whether it opens upward (a > 0) or downward (a < 0). For a quadratic equation, a must be non-zero.
• Coefficient b: The number multiplying x. It affects the horizontal position of the parabola and the location of the vertex.
• Coefficient c: The constant term. It determines where the graph crosses the y-axis (the y-intercept).
• Quadratic equation: An equation that can be written in the form ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola.
• Root / solution: A value of x that makes the equation true. In other words, plugging the root into ax² + bx + c gives 0.
• Discriminant (D): The expression b² − 4ac that tells you how many and what kind of roots the equation has.
• Real root: A solution that is a real number and can be plotted on the usual number line.
• Complex root: A solution that includes an imaginary part involving i, where i² = −1. Complex roots often appear in conjugate pairs.

Formulas used in the Quadratic Equation Solver

This solver is based on the standard quadratic formula, which you can use for any equation of the form ax² + bx + c = 0 with a ≠ 0.

1. Discriminant

The discriminant tells you how many roots there are and whether they are real or complex:
D = b² − 4ac

• D > 0 → two distinct real roots
• D = 0 → one real root (a double root)
• D < 0 → two complex conjugate roots

2. Quadratic formula for the roots

For a quadratic equation ax² + bx + c = 0 with a ≠ 0, the roots are given by:
x = [-b ± √(b² − 4ac)] ÷ (2a)

When D > 0, √D is real and you get two distinct real roots:
x₁ = (-b + √D) ÷ (2a), x₂ = (-b − √D) ÷ (2a).

When D = 0, √D = 0, so both roots are the same:
x₁ = x₂ = -b ÷ (2a).

When D < 0, √D is imaginary. Write D = −K with K > 0, then:
x = [-b ± i√K] ÷ (2a)
The real part is -b ÷ (2a) and the imaginary part is √K ÷ (2a).

These formulas are standard in algebra courses and are widely used in physics, engineering, finance, and any situation where quadratic relationships appear.

Quadratic Equation Solver FAQs

• What is a quadratic equation in simple terms?
  A quadratic equation is any equation that can be written in the form ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola. Quadratic equations appear in physics, geometry, finance, and everyday word problems involving areas, projectiles, and optimization.
• What does the discriminant tell me about the roots?
  The discriminant D = b² − 4ac tells you how many and what type of roots you have. If D > 0, there are two distinct real roots. If D = 0, there is one real root with multiplicity 2. If D < 0, there are two complex conjugate roots.
• What happens if the coefficient a is zero?
  If a = 0, the equation is no longer quadratic. It becomes linear (bx + c = 0) if b ≠ 0, or not a meaningful equation if both a and b are zero. This solver focuses on true quadratic equations where a is non-zero, but it will note when your input describes a linear case.
• How are complex roots shown in this calculator?
  When the discriminant is negative, the roots are complex. This calculator writes them in the form x = p ± qi, where p is the real part and q is the coefficient of the imaginary unit i. The two solutions are complex conjugates of each other.
• Is this quadratic solver accurate enough for homework and exams?
  The calculator uses standard floating-point arithmetic and shows roots to several decimal places. For most homework, quizzes, and exams, you can round to two or three decimal places as required by your teacher. It is always a good idea to plug the solutions back into the original equation to verify the result.
• When should I use factoring instead of the quadratic formula?
  Factoring is great when the quadratic has “nice” integer roots and factors cleanly. The quadratic formula, however, works for every quadratic equation, even when the roots are irrational or complex. Many students use the quadratic formula as a reliable fallback when factoring is difficult or impossible.