🎯 Algebra Solver

Quadratic Formula Calculator

Q: What is the quadratic formula?

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), and it is used to solve quadratic equations written in the form ax² + bx + c = 0, where a ≠ 0.

Q: How do you use the quadratic formula?

First write the equation in standard form ax² + bx + c = 0. Then identify a, b, and c, calculate the discriminant b² - 4ac, substitute the values into the quadratic formula, and calculate both roots.

Q: What does the discriminant tell you?

The discriminant Δ = b² - 4ac determines the type of roots. If Δ is positive there are two distinct real roots, if Δ is zero there is one repeated real root, and if Δ is negative there are two complex roots.

Q: Can the quadratic formula give complex roots?

Yes. When the discriminant is negative, the square root contains a negative number and the equation has two complex conjugate roots involving the imaginary unit i.

Q: When should you use the quadratic formula?

You can use the quadratic formula to solve any quadratic equation with a ≠ 0. It is especially useful when factoring is difficult or does not give an obvious solution.

Solve quadratic equations using the quadratic formula with step-by-step solutions, discriminant, and real or complex roots.

Quadratic Coefficients (ax² + bx + c = 0)

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Coefficient (a) ℹ️ Info

Coefficient (b) ℹ️ Info

Constant (c) ℹ️ Info

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⚡ Quick Tools: 🧩 Quadratic Factoring Calculator 🔍 Discriminant Calculator (Δ = b² - 4ac) 🔲 Completing the Square Calculator

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Calculation Reference Table

Benchmark values and quick reference intervals

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How to Use & Educational Guide

In-depth guide for Quadratic Formula Solver

Educational Overview

What Is the Quadratic Formula?

The quadratic formula is a standard method for solving any quadratic equation written in the form $ax^2 + bx + c = 0$, where $a \ne 0$. It can be used to find the roots or solutions of a quadratic equation when factoring is difficult or does not work with simple integers. The formula can produce two real roots, one repeated real root, or two complex roots.

The Quadratic Formula and Discriminant

For a quadratic equation in standard form $ax^2 + bx + c = 0$, use the following formula to solve for $x$:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The expression under the square root is called the discriminant:

$$\Delta = b^2 - 4ac$$

Δ > 0: Two distinct real roots.

Δ = 0: One repeated real root.

Δ < 0: Two complex roots.

How to Use the Quadratic Formula Step-by-Step?

To solve a quadratic equation using the formula, follow these steps:

Write the equation in standard form: Rearrange the equation so it becomes $ax^2 + bx + c = 0$.
Identify $a$, $b$, and $c$: Read the coefficients directly from the standard form, including their signs.
Calculate the discriminant: Find $\Delta = b^2 - 4ac$.
Substitute into the quadratic formula: Replace $a$, $b$, and $c$ in $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
Calculate both solutions: Use the plus and minus signs separately to find $x_1$ and $x_2$.
Simplify and verify: Simplify the roots and substitute them into the original equation when necessary.

Quadratic Formula Examples

Example 1: Two Real Roots — $x^2 - 5x + 6 = 0$

• Identify coefficients: $a = 1$, $b = -5$, $c = 6$.

• Discriminant: $\Delta = (-5)^2 - 4(1)(6) = 25 - 24 = 1$.

• Substitute: $x = \frac{-(-5) \pm \sqrt{1}}{2(1)} = \frac{5 \pm 1}{2}$.

• First root: $x_1 = \frac{5 + 1}{2} = 3$.

• Second root: $x_2 = \frac{5 - 1}{2} = 2$.

👉 Answer: $x_1 = 3,\; x_2 = 2$

Example 2: One Repeated Root — $x^2 - 6x + 9 = 0$

• Coefficients: $a = 1$, $b = -6$, $c = 9$.

• Discriminant: $\Delta = (-6)^2 - 4(1)(9) = 36 - 36 = 0$.

• Substitute: $x = \frac{-(-6) \pm \sqrt{0}}{2} = \frac{6}{2}$.

👉 Answer: $x = 3$ (repeated root)

Example 3: Complex Roots — $x^2 + 4x + 5 = 0$

• Coefficients: $a = 1$, $b = 4$, $c = 5$.

• Discriminant: $\Delta = 4^2 - 4(1)(5) = 16 - 20 = -4$.

• Square root: $\sqrt{-4} = 2i$.

• Substitute: $x = \frac{-4 \pm 2i}{2}$.

👉 Complex Roots: $x_1 = -2 + i,\; x_2 = -2 - i$

Example 4: Quadratic Equation Not Initially in Standard Form — $2x^2 + 3 = 7x$

• Step 1: Move all terms to one side: $2x^2 - 7x + 3 = 0$.

• Step 2: Identify $a = 2$, $b = -7$, $c = 3$.

• Step 3: Calculate $\Delta = (-7)^2 - 4(2)(3) = 49 - 24 = 25$.

• Step 4: $x = \frac{7 \pm \sqrt{25}}{4} = \frac{7 \pm 5}{4}$.

• Roots: $x_1 = 3$ and $x_2 = \frac{1}{2}$.

👉 Answer: $x = 3,\; \frac{1}{2}$

When Should You Use the Quadratic Formula?

The quadratic formula is useful whenever you need to solve a quadratic equation and factoring is inconvenient, difficult, or does not produce obvious integer factors. It provides a reliable method for finding both roots of any quadratic equation with $a \ne 0$.

📐 Finding Parabola Intercepts

For a quadratic function $y = ax^2 + bx + c$, solving $ax^2 + bx + c = 0$ gives the x-intercepts of the parabola when real roots exist.

🚀 Physics and Motion

Quadratic equations can model projectile motion. Setting a height equation equal to zero can help determine when an object reaches the ground.

Common Quadratic Formula Mistakes

Not using standard form: Always rearrange the equation into $ax^2 + bx + c = 0$ before identifying $a$, $b$, and $c$.
Missing the negative sign: If $b = -5$, then $-b = -(-5) = 5$.
Forgetting the entire numerator: Both $-b$ and $\pm\sqrt{b^2-4ac}$ belong in the numerator and are divided by $2a$.
Incorrectly squaring negative values: When $b = -5$, calculate $b^2 = (-5)^2 = 25$.
Ignoring the discriminant: A negative discriminant means the equation has complex roots rather than real roots.

Solve a Quadratic Equation with the Calculator

Enter the coefficients of your quadratic equation into the Quadratic Formula Calculator to solve for $x$. The calculator evaluates the discriminant, applies the quadratic formula, and shows the roots with step-by-step working.

Mathematical accuracy verified by AptCalc Engine

Frequently Asked Questions (FAQs)

Answers to common questions about this calculator

What is the quadratic formula?

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a), and it is used to solve quadratic equations written in the form ax² + bx + c = 0, where a ≠ 0.

How do you use the quadratic formula?

First write the equation in standard form ax² + bx + c = 0. Then identify a, b, and c, calculate the discriminant b² - 4ac, substitute the values into the quadratic formula, and calculate both roots.

What does the discriminant tell you?

The discriminant Δ = b² - 4ac determines the type of roots. If Δ is positive there are two distinct real roots, if Δ is zero there is one repeated real root, and if Δ is negative there are two complex roots.

Can the quadratic formula give complex roots?

Yes. When the discriminant is negative, the square root contains a negative number and the equation has two complex conjugate roots involving the imaginary unit i.

When should you use the quadratic formula?

You can use the quadratic formula to solve any quadratic equation with a ≠ 0. It is especially useful when factoring is difficult or does not give an obvious solution.