Quadratic Formula Calculator with Steps

This quadratic formula calculator solves any quadratic equation of the form ax² + bx + c = 0 and shows every step of the solution. You enter the coefficients a, b, and c, and the calculator returns the real or complex roots, the discriminant, and a clear breakdown of how the formula was applied.

Formula

The quadratic formula is the standard method for finding the roots of a quadratic equation. For an equation written as:

ax² + bx + c = 0

the roots are given by:

x = (−b ± √(b² − 4ac)) / (2a)

Here, a, b, and c are real numbers (with a ≠ 0), and x represents the unknown variable. The expression under the square root, b² − 4ac, is called the discriminant (often written as Δ). It determines the nature of the roots:

• If Δ > 0, there are two distinct real roots.
• If Δ = 0, there is exactly one real root (a repeated root).
• If Δ < 0, there are two complex conjugate roots.

The formula works for any quadratic equation, provided you first rearrange it into standard form (all terms on one side, equal to zero).

Worked Example

Let’s solve the equation x² − 3x + 2 = 0 using the default values: a = 1, b = −3, c = 2.

1. Identify the coefficients: a = 1, b = −3, c = 2.
2. Calculate the discriminant: Δ = b² − 4ac = (−3)² − 4(1)(2) = 9 − 8 = 1.
3. Apply the quadratic formula: x = (−b ± √Δ) / (2a) = (3 ± √1) / (2×1).
4. Simplify: √1 = 1, so x = (3 ± 1) / 2.
5. Find the two roots: x� = (3 + 1)/2 = 2, x₂ = (3 − 1)/2 = 1.

The calculator returns x� = 2 and x₂ = 1, matching the steps above. You can verify by plugging either root back into the original equation: 1² − 3(1) + 2 = 0 and 2² − 3(2) + 2 = 0.

When to Use the Quadratic Formula

The quadratic formula is the most general method for solving quadratic equations. It works for every quadratic, even when factoring is difficult or impossible. You should use it when:

• The equation cannot be easily factored (e.g., x² + 2x + 5 = 0).
• You need exact answers, including complex roots.
• You are solving physics or geometry problems involving projectile motion, area, or optimization.
• You want a reliable check for answers obtained by factoring or completing the square.

Compared to factoring, the formula is more systematic. Compared to completing the square, it is faster and less prone to algebraic errors. For simple equations like x² − 5x + 6 = 0, factoring may be quicker, but the formula always works.

How to Read the Results

The calculator shows two roots, x� and x₂, and the discriminant. Here’s how to interpret them:

• Real roots: If the discriminant is positive, you get two different real numbers. These are the x-values where the parabola crosses the x-axis.
• One root: If the discriminant is zero, both roots are the same. The parabola touches the x-axis at exactly one point (the vertex).
• Complex roots: If the discriminant is negative, the roots involve the imaginary unit i (where i² = −1). For example, x = 1 ± 2i. The parabola does not intersect the x-axis at all.

In practical problems, real roots usually have a physical meaning (like time or distance), while complex roots often indicate that no real solution exists in the given context.

Common Mistakes and How to Avoid Them

Several errors are common when applying the quadratic formula manually:

• Sign errors: Forgetting that −b changes the sign of b. If b = −3, then −b = 3, not −3.
• Incorrect a, b, c: The equation must be in standard form. For 2x² + 5x = 3, you must subtract 3 from both sides to get 2x² + 5x − 3 = 0, so c = −3.
• Misplacing the denominator: The entire numerator (−b ± √Δ) is divided by 2a, not just the square root part.
• Rounding too early: If you round the discriminant before taking the square root, you may get a noticeably different answer. Keep full precision until the final step.
• Ignoring complex roots: When the discriminant is negative, the equation still has solutions, just not real ones. The calculator shows them as complex numbers.

Limitations of the Formula

The quadratic formula only applies to polynomial equations of degree 2. It cannot solve cubic or higher-degree equations directly. Also, if a = 0, the equation is not quadratic, and the formula is undefined (division by zero). In that case, you have a linear equation and should solve it differently.

Another limitation is numerical precision. For very large or very small coefficients, the standard formula can suffer from cancellation errors when using floating-point arithmetic. In such cases, an alternative form using the sum and product of roots may be more stable, but for most everyday problems the standard formula is accurate enough.

Frequently Asked Questions

What if the discriminant is negative?

If the discriminant is negative, the quadratic equation has no real solutions, but it has two complex solutions. For example, x² + 4x + 5 = 0 has discriminant 16 − 20 = −4, giving roots x = −2 ± i. The calculator will show these as complex numbers with an imaginary part.

Can I use the quadratic formula for any quadratic equation?

Yes, as long as the equation is a quadratic (degree 2) and a ≠ 0. You must first rearrange it into the standard form ax² + bx + c = 0. Then you can plug in the coefficients and get the roots.

How do I find a, b, and c from an equation?

Write the equation in the form ax² + bx + c = 0. The coefficient of x² is a, the coefficient of x is b, and the constant term is c. For example, in 3x² − 2x + 7 = 0, a = 3, b = −2, c = 7. If a term is missing, its coefficient is 0 (e.g., x² − 4 = 0 has b = 0).

What is the difference between real and complex roots?

Real roots are numbers that can be plotted on the number line (like 2 or −0.5). Complex roots include the imaginary unit i and appear as a + bi. They occur when the discriminant is negative, meaning the parabola never touches the x-axis. Both are valid solutions to the equation.

Why does the calculator show two roots?

A quadratic equation always has two roots (counting multiplicity). The ± symbol in the formula gives two possibilities: one with addition and one with subtraction. Even when the discriminant is zero, the two roots are equal, so there is effectively one distinct root.

Can the quadratic formula be derived?

Yes, it comes from completing the square on the general equation ax² + bx + c = 0. By dividing by a, moving the constant, and adding (b/2a)², you can isolate x and arrive at the formula. This derivation is a standard algebra exercise.

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in a | Coefficient a | - | 1 | 0
in b | Coefficient b | - | -3 | 0
in c | Coefficient c | - | 2 | 0
out x1 | First root | x� | (-b + sqrt(b^2 - 4*a*c)) / (2*a) | 4
out x2 | Second root | x₂ | (-b - sqrt(b^2 - 4*a*c)) / (2*a) | 4
out disc | Discriminant | Δ | b^2 - 4*a*c | 0