Quadratic Formula Calculator

Introduction to the quadratic formula calculator

This quadratic formula calculator solves equations written in the standard form ax² + bx + c = 0. Enter the coefficients a, b, and c, and it returns the roots of x along with a discriminant check. That check explains whether the equation has two real answers, one repeated real answer, or a pair of complex answers.

Quadratic equations describe curved, parabolic relationships. They appear in algebra courses, projectile motion, optimization problems, and curved cost or revenue models. The quadratic formula is especially useful when factoring is difficult or impossible over the integers, because it works for every genuine quadratic with a ≠ 0.

If a = 0, the equation is no longer quadratic. The calculator then treats it as a linear equation when b ≠ 0. If both a and b are zero, the result depends on c: there is no solution when c ≠ 0, and every real number works when c = 0.

How to use the quadratic formula calculator

To use this quadratic formula calculator accurately, first arrange the equation so that every term is on one side and the other side is zero. Then read the coefficient attached to each power of x rather than copying the visible signs casually.

1. Rewrite your equation in the form ax² + bx + c = 0, moving every term to one side.
2. Enter the numerical values of a, b, and c. Negative numbers and decimals are accepted.
3. Select Solve equation to calculate the roots.
4. Read the result area to identify two real roots, a repeated root, a complex pair, or a linear fallback.

For example, 2x² = 8x − 6 becomes 2x² − 8x + 6 = 0, so enter a = 2, b = -8, and c = 6. Likewise, -x² + 5 = 2x becomes -x² − 2x + 5 = 0. A negative leading coefficient is valid; the formula still works normally.

The quadratic formula and discriminant assumptions

For a true quadratic equation, this calculator applies the standard quadratic formula. The plus-or-minus sign means that one calculation produces two possible roots whenever the square-root term is nonzero.

The expression under the square root is the discriminant, written as Δ. In this quadratic formula calculator, the sign of Δ determines the kind of solution before the roots are displayed.

• Δ > 0: two distinct real roots.
• Δ = 0: one repeated real root, also called a double root.
• Δ < 0: two complex roots forming a conjugate pair.

When a = 0, the remaining equation is bx + c = 0, so its solution is x = -c / b if b ≠ 0. When the discriminant is negative, the calculator expresses the roots in p ± qi form. Here i is the imaginary unit, defined by i² = -1.

Worked examples for quadratic roots and the discriminant

These quadratic formula calculator examples show the three main discriminant outcomes. They also illustrate why the signs of the coefficients matter as much as the arithmetic itself.

Example 1: two real roots

Solve x² − 3x + 2 = 0. The coefficients are a = 1, b = -3, and c = 2. The discriminant is (-3)² − 4(1)(2) = 1, which is positive. The roots are (3 ± √1) / 2, giving x = 2 and x = 1. Entering those coefficients should display two real roots close to 2.000000 and 1.000000.

Example 2: a repeated root on the x-axis

Solve 2x² + 8x + 8 = 0. Here a = 2, b = 8, and c = 8. The discriminant is 8² − 4(2)(8) = 0, so the two formula branches meet at the same result: x = -8 / 4 = -2. On a graph, this means the parabola touches the x-axis at one point and turns around instead of crossing it.

Example 3: complex conjugate roots

Solve x² + 2x + 5 = 0. With a = 1, b = 2, and c = 5, the discriminant is 2² − 4(1)(5) = -16. Because it is negative, there are no real x-intercepts. The real part is -2 / 2 = -1, and the imaginary magnitude is √16 / 2 = 2, so the roots are x = -1 ± 2i. Complex roots are useful in intermediate mathematical models, including oscillations and control systems, even when a final real-world quantity must be real.

Common quadratic coefficient mistakes and sign checks

Most incorrect quadratic formula calculator results come from entering the wrong coefficients, not from the formula. A quick rewrite and sign check prevents many of these errors.

• Move every term first: x² + 4x = 1 becomes x² + 4x - 1 = 0, so c is -1.
• Distribute a minus sign: x² - (3x - 2) = 0 becomes x² - 3x + 2 = 0.
• Keep an implied coefficient: in x² - x + 6 = 0, the middle coefficient is b = -1, not 1.
• Handle fractions consistently: enter 0.5 and 0.75, or multiply the whole equation by a common denominator. Both methods preserve the roots.

After solving, substitute a displayed root into ax² + bx + c. A value close to zero is a useful check, allowing for the calculator’s displayed rounding.

Understanding quadratic formula calculator output

The quadratic formula calculator result area uses concise labels so the solution type is immediately clear. Two real roots are two different numbers, while one repeated real root is a single number that occurs twice algebraically. Complex roots share the same real part and have imaginary parts with opposite signs.

The calculator also identifies linear cases when a = 0. If both a and b are zero, it distinguishes an inconsistent equation from an identity that is true for all real numbers. Displayed numeric values are rounded to six decimal places, so irrational or repeating answers are approximations. Multiplying all three coefficients by the same nonzero factor changes the equation’s scale but does not change its roots.

Limitations of this quadratic formula calculation

This quadratic formula calculator is designed for numerical root finding, so a few practical limits matter. JavaScript floating-point arithmetic can introduce small rounding effects with extremely large or tiny coefficients. Similarly, a discriminant extremely close to zero may be affected by floating-point precision when it is classified as positive, zero, or negative.

The page does not simplify radicals into exact symbolic forms and does not graph the parabola. It reports complex answers in decimal a ± bi form. Finally, a mathematical root is not automatically a meaningful answer in every application: a negative root may be impossible when x represents elapsed time, distance, or another quantity restricted to nonnegative values.

Frequently asked questions about the quadratic formula calculator

These quadratic formula calculator questions address the most common points that arise when entering coefficients and interpreting roots.

Does the quadratic formula always work?

Yes, for every quadratic equation with a ≠ 0. This calculator uses the formula rather than factoring, so it can find numerical roots even when the expression does not factor neatly. When a = 0, it uses the appropriate linear fallback when one exists.

Why did I get complex roots?

Complex roots occur when the discriminant is negative. Recheck the sign of c after moving all terms to one side, and make sure b and c were not swapped during entry.

What does a repeated root mean?

A repeated root occurs when Δ = 0. The parabola meets the x-axis once, and the quadratic can be written in the form a(x - r)², where r is that repeated root.

Can I enter integers, decimals, and negative values?

Yes. The inputs accept real numeric values, including negatives and decimals. Convert fractions to decimals or multiply the complete equation by a common denominator before entering the coefficients.