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Quadratic Formula Calculator

Q: How do I interpret complex solutions in a real-world context?

Complex solutions mean that the parabola never crosses the x-axis, so there are no real values of x satisfying the equation. In physics or engineering, this often signals that a problem has no physical solution under the given constraints. For instance, if a projectile motion equation yields complex roots, it means the projectile never reaches a target height. While complex solutions are mathematically valid, they usually indicate an infeasible scenario in applied contexts.

Q: What if my equation doesn't look like Ax² + Bx + C = 0?

Rearrange it into standard form by moving all terms to one side and combining like terms. For example, 2x(x + 3) = 10 becomes 2x² + 6x − 10 = 0 , then x² + 3x − 5 = 0 (optionally dividing by 2). You can also expand, distribute, and simplify fractional coefficients. Once in standard form, identify A , B , and C carefully, including their signs.

Quadratic equations appear everywhere—from physics and engineering to finance and design. When you need to solve equations in the standard form Ax² + Bx + C = 0, the quadratic formula is your universal tool. Unlike factoring (which works only for special cases), the formula works for any quadratic equation. Simply enter your coefficients and discover whether your equation has two real solutions, one repeated root, or complex solutions.

Last updated: May 6, 2026

Creators Mateusz Mucha, BEng

Reviewers Anna Szczepanek and Jasmine J. Mah

3,830 people find this calculator helpful

The Quadratic Formula

Any equation of the form Ax² + Bx + C = 0 can be solved using the quadratic formula. The key insight is that the discriminant—the expression under the square root—tells you exactly how many and what type of solutions exist.

x = (−B ± √Δ) / 2A

where Δ = B² − 4AC

A — Coefficient of x² (cannot be zero, or the equation is linear, not quadratic)
B — Coefficient of x
C — Constant term
Δ (discriminant) — Determines the nature and number of solutions: positive means two real roots, zero means one repeated root, negative means two complex conjugate roots

Understanding the Discriminant

The discriminant is the expression B² − 4AC that appears under the square root in the quadratic formula. Its value tells you everything about the solutions before you even calculate them:

Δ > 0: The equation has two distinct real solutions. For example, x² − 5x + 6 = 0 has discriminant 25 − 24 = 1, yielding roots x = 2 and x = 3.
Δ = 0: The equation has exactly one real solution (a repeated root). This parabola just touches the x-axis. For instance, x² − 2x + 1 = 0 has discriminant 4 − 4 = 0 and one solution x = 1.
Δ < 0: The equation has no real solutions; instead, there are two complex conjugate roots. The parabola never crosses the x-axis. For example, x² + x + 1 = 0 has discriminant 1 − 4 = −3, giving complex solutions involving the imaginary unit i.

How to Use the Formula Step-by-Step

To solve 3x² − 7x + 2 = 0 using the quadratic formula, follow these steps:

Identify your coefficients: A = 3, B = −7, C = 2.
Calculate the discriminant: Δ = (−7)² − 4(3)(2) = 49 − 24 = 25.
Check the sign: Since Δ = 25 > 0, you have two real solutions.
Apply the formula: x = (7 ± √25) / 6 = (7 ± 5) / 6, giving x₁ = 2 and x₂ = 1/3.

Always first rearrange your equation into standard form Ax² + Bx + C = 0 before identifying coefficients. Move all terms to one side and combine like terms.

Complex Solutions and the Imaginary Unit

When the discriminant is negative, you cannot take a real square root, but you can work with complex numbers. Complex solutions are written as p + qi, where p is the real part, q is the imaginary part, and i = √(−1).

For an equation with a negative discriminant, the two solutions are always complex conjugates: if one solution is p + qi, the other is p − qi. For example, solving x² + 2x + 5 = 0 yields Δ = 4 − 20 = −16. The real part is −2/2 = −1, and the imaginary part is ±√16/2 = ±2, giving solutions −1 + 2i and −1 − 2i.

Common Pitfalls and Practical Tips

Avoid these frequent mistakes when applying the quadratic formula.

Don't forget to rearrange first — Your equation must be in the form <code>Ax² + Bx + C = 0</code> with zero on the right side. If you see <code>3x² + x = 5</code>, rewrite it as <code>3x² + x − 5 = 0</code> before identifying coefficients. Errors at this stage propagate through the entire calculation.
Watch the signs of your coefficients — Include the sign as part of each coefficient. In <code>2x² − 3x − 4 = 0</code>, you have <code>A = 2</code>, <code>B = −3</code>, and <code>C = −4</code>, not <code>B = 3</code> and <code>C = 4</code>. Incorrect signs in the formula lead to completely wrong roots.
Check whether A equals zero — If <code>A = 0</code>, you don't have a quadratic equation—you have a linear equation. The quadratic formula will fail because you'll divide by zero. For example, <code>0x² + 5x + 3 = 0</code> simplifies to <code>5x + 3 = 0</code>, solved as <code>x = −3/5</code> using basic algebra.
Handle complex solutions carefully — When Δ < 0, ensure you correctly compute the imaginary part as <code>±√|Δ| / 2A</code> (using the absolute value of Δ). Many calculators handle this automatically, but if calculating by hand, remember that <code>√(−16) = 4i</code>, not <code>−4i</code>.

Frequently Asked Questions

What happens if the discriminant equals zero?

When the discriminant is zero, the quadratic equation has exactly one real solution (called a repeated or double root). Geometrically, the parabola just touches the x-axis at one point. Using the quadratic formula, both the plus and minus branches give the same result: <code>x = −B / 2A</code>. For instance, <code>x² − 4x + 4 = 0</code> has Δ = 0 and the single solution <code>x = 2</code>.

Can the quadratic formula solve all quadratic equations?

Yes, the quadratic formula is universal for any equation of the form <code>Ax² + Bx + C = 0</code> where <code>A ≠ 0</code>. Unlike factoring or completing the square, it always works regardless of whether coefficients are integers, fractions, or decimals. It also handles cases where real solutions don't exist, finding complex solutions instead. This makes it the most reliable method for solving quadratics.

Why does the quadratic formula have a plus-minus symbol?

The ± symbol indicates that the quadratic formula yields two values (in the general case): one by adding the square root and one by subtracting it. This reflects the fact that quadratics typically have two solutions. When Δ > 0, these are two distinct real numbers. When Δ = 0, both branches give the same number. When Δ < 0, you get two complex conjugates that differ only in the sign of their imaginary parts.

How do I interpret complex solutions in a real-world context?

Complex solutions mean that the parabola never crosses the x-axis, so there are no real values of x satisfying the equation. In physics or engineering, this often signals that a problem has no physical solution under the given constraints. For instance, if a projectile motion equation yields complex roots, it means the projectile never reaches a target height. While complex solutions are mathematically valid, they usually indicate an infeasible scenario in applied contexts.

What if my equation doesn't look like Ax² + Bx + C = 0?

Rearrange it into standard form by moving all terms to one side and combining like terms. For example, <code>2x(x + 3) = 10</code> becomes <code>2x² + 6x − 10 = 0</code>, then <code>x² + 3x − 5 = 0</code> (optionally dividing by 2). You can also expand, distribute, and simplify fractional coefficients. Once in standard form, identify <code>A</code>, <code>B</code>, and <code>C</code> carefully, including their signs.

Is the quadratic formula the only way to solve quadratic equations?

No, there are alternative methods: factoring works when the trinomial factors neatly; completing the square is useful for understanding the vertex form and is more intuitive for some; graphing gives a visual solution. However, the quadratic formula is fastest and most reliable, especially for equations with awkward coefficients or irrational solutions. Most professionals and calculators use the formula as the default method.

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Select formula and enter parameters

Formula form

x₁

x₂

x₁,₂ real part

x₁,₂ imaginary part +/−

Results (Δ & solutions)

Discriminant (Δ)