Quadratic Formula Calculator

What is the Quadratic Formula?

The quadratic formula is a fundamental mathematical tool used to find the solutions (roots) of quadratic equations. A quadratic equation is any polynomial equation of the second degree, written in the standard form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The quadratic formula provides an algebraic method to solve for the unknown variable x without requiring factorization or completing the square.

The formula itself is expressed as: x = [-b ± √(b² - 4ac)] / 2a. This elegant equation was derived through the method of completing the square and works for all quadratic equations, regardless of whether the roots are real, repeated, or complex. The ± symbol indicates that there are typically two solutions to a quadratic equation, obtained by using both the positive and negative values of the square root term.

The expression under the square root, b² - 4ac, is called the discriminant and plays a crucial role in determining the nature of the solutions. When the discriminant is positive, the equation has two distinct real roots. When it equals zero, there is exactly one real root (a repeated root). When the discriminant is negative, the equation has two complex conjugate roots involving the imaginary unit i, where i = √(-1).

Our advanced quadratic formula calculator not only computes the roots but also provides step-by-step solutions, discriminant analysis, and both exact (simplified radical) and decimal approximations. Whether you're a student learning algebra, an engineer solving physics problems, or a professional working with parabolic models, this tool offers accurate and comprehensive solutions for all your quadratic equation needs.

Understanding the Discriminant (b² - 4ac)

The discriminant is the key to understanding what kind of solutions a quadratic equation will have. Here's a comprehensive guide to interpreting the discriminant:

Discriminant > 0 (Positive)

When b² - 4ac > 0, the quadratic equation has two distinct real roots. This means the parabola crosses the x-axis at two different points.

Example: x² - 5x + 6 = 0
Discriminant = 25 - 24 = 1 > 0
Roots: x₁ = 3, x₂ = 2

Discriminant = 0 (Zero)

When b² - 4ac = 0, the quadratic equation has one repeated real root (also called a double root). The parabola just touches the x-axis at exactly one point (the vertex).

Example: x² - 4x + 4 = 0
Discriminant = 16 - 16 = 0
Root: x = 2 (repeated)

Discriminant < 0 (Negative)

When b² - 4ac < 0, the quadratic equation has two complex conjugate roots. These roots involve the imaginary unit i, and the parabola does not intersect the x-axis.

Example: x² + 2x + 5 = 0
Discriminant = 4 - 20 = -16 < 0
Roots: x₁ = -1 + 2i, x₂ = -1 - 2i

How to Use the Quadratic Formula Calculator

Identify your coefficients: Write your quadratic equation in standard form ax² + bx + c = 0 and identify the values of a, b, and c.
Enter the values: Input the coefficients a, b, and c into the calculator. Remember that a cannot be zero (otherwise it's not a quadratic equation).
Click "Solve Equation": The calculator will compute the discriminant, apply the quadratic formula, and determine the type and value of the roots.
Review the solution: Examine the step-by-step solution including discriminant calculation, formula application, and both exact and decimal forms of the roots.
Verify your answer: You can substitute the roots back into the original equation to verify they satisfy ax² + bx + c = 0.

Step-by-Step Examples

Example 1: Two Real Roots

Solve: x² - 7x + 10 = 0

Step 1: Identify coefficients: a = 1, b = -7, c = 10

Step 2: Calculate discriminant: b² - 4ac = (-7)² - 4(1)(10) = 49 - 40 = 9

Step 3: Since 9 > 0, we have two real roots

Step 4: Apply formula: x = [7 ± √9] / 2 = [7 ± 3] / 2

Step 5: Solutions:

x₁ = (7 + 3) / 2 = 10 / 2 = 5

x₂ = (7 - 3) / 2 = 4 / 2 = 2

Example 2: One Repeated Root

Solve: 4x² - 12x + 9 = 0

Step 1: Identify coefficients: a = 4, b = -12, c = 9

Step 2: Calculate discriminant: b² - 4ac = (-12)² - 4(4)(9) = 144 - 144 = 0

Step 3: Since discriminant = 0, we have one repeated root

Step 4: Apply formula: x = -(-12) / (2 × 4) = 12 / 8 = 3/2

Step 5: Solution: x = 3/2 or 1.5

Example 3: Complex Roots

Solve: x² + 4x + 13 = 0

Step 1: Identify coefficients: a = 1, b = 4, c = 13

Step 2: Calculate discriminant: b² - 4ac = (4)² - 4(1)(13) = 16 - 52 = -36

Step 3: Since -36 < 0, we have two complex roots

Step 4: Apply formula: x = [-4 ± √(-36)] / 2 = [-4 ± 6i] / 2

Step 5: Solutions:

x₁ = (-4 + 6i) / 2 = -2 + 3i

x₂ = (-4 - 6i) / 2 = -2 - 3i

Why Use Our Quadratic Formula Calculator?

✓Step-by-Step Solutions

See detailed working for every calculation including discriminant analysis and formula application, perfect for learning and verification.

✓Exact & Decimal Forms

Get both simplified radical/fractional forms and decimal approximations for maximum flexibility in your work.

✓Complex Number Support

Handles all types of roots including complex numbers with imaginary components, displayed in standard a + bi form.

✓Mobile Optimized

Fully responsive design works perfectly on all devices from smartphones to desktop computers.

✓Instant Results

Fast calculations with immediate display of roots, discriminant, and complete solution steps.

✓100% Free

No registration, no fees, no limitations. Use our calculator as many times as you need, completely free.

Real-World Applications of Quadratic Equations

📐 Physics and Engineering

Quadratic equations model projectile motion, where the height of an object follows a parabolic path. The equation h(t) = -16t² + v₀t + h₀ describes the height of a projectile at time t, where v₀ is initial velocity and h₀ is initial height. Engineers use quadratic formulas to calculate optimal launch angles, maximum heights, and impact times.

Example: A ball thrown upward at 48 ft/s from 6 feet high reaches the ground when -16t² + 48t + 6 = 0.

💰 Business and Economics

Revenue and profit optimization often involve quadratic functions. The profit function P(x) = -ax² + bx - c represents how profit changes with production quantity. Business analysts use the quadratic formula to find break-even points (where P(x) = 0) and maximum profit levels by analyzing the vertex of the parabola.

Example: If profit P(x) = -2x² + 80x - 600, solving P(x) = 0 gives break-even production quantities.

🏗️ Architecture and Design

Parabolic arches and curves in architecture follow quadratic equations. Designers use these formulas to calculate structural dimensions, ensure proper load distribution, and create aesthetically pleasing curves. Suspension bridges, satellite dishes, and reflector designs all rely on quadratic mathematics.

Example: A parabolic arch with equation y = -0.1x² + 4x must satisfy specific height requirements at various points.

🌾 Agriculture and Land Management

Farmers and land planners use quadratic equations to optimize rectangular field areas with fixed perimeter constraints. Given perimeter P, if length = x, then width = (P/2 - x), and area A = x(P/2 - x) = -x² + (P/2)x, a quadratic function maximized at the vertex.

Example: With 200m of fencing, find dimensions that maximize area: A = -x² + 100x.

🎮 Computer Graphics and Gaming

Video game physics engines use quadratic equations to simulate realistic motion, collision detection, and trajectory calculations. The parabolic paths of jumping characters, flying projectiles, and falling objects are all governed by quadratic mathematics.

Example: A character's jump arc y = -0.5x² + 3x models the height versus horizontal distance traveled.

🔬 Science and Research

Chemical reaction rates, population growth models, and orbital mechanics often involve quadratic relationships. Scientists use the quadratic formula to solve for equilibrium points, critical concentrations, and time parameters in experimental data.

Example: A population model P(t) = 5t² - 30t + 100 might need solving for specific population levels.

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2, written in the standard form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The term "quadratic" comes from "quadratus," the Latin word for square, because the variable is squared (raised to the power of 2).

When should I use the quadratic formula?

Use the quadratic formula when you need to solve any quadratic equation, especially when factoring is difficult or impossible. It's particularly useful for equations with decimal coefficients, large numbers, or when you need exact radical forms. The formula works for all quadratic equations, making it the most universal solution method.

What does the discriminant tell us?

The discriminant (b² - 4ac) determines the nature of the roots: If positive, there are two distinct real roots; if zero, one repeated real root; if negative, two complex conjugate roots. The discriminant also indicates whether the parabola crosses the x-axis (positive), touches it (zero), or doesn't intersect it (negative).

Can the coefficient 'a' be zero?

No, if a = 0, the equation becomes bx + c = 0, which is a linear equation, not a quadratic equation. The coefficient 'a' must be non-zero for the equation to be quadratic (degree 2). Our calculator will show an error if you try to enter a = 0.

What are complex roots?

Complex roots occur when the discriminant is negative, requiring the square root of a negative number. These roots are expressed in the form a + bi, where i = √(-1) is the imaginary unit. Complex roots always come in conjugate pairs: if a + bi is a root, then a - bi is also a root.

How do I verify my solution?

Substitute each root back into the original equation ax² + bx + c = 0. If the root is correct, the left side should equal zero. For example, if x = 3 is a root of x² - 5x + 6 = 0, then (3)² - 5(3) + 6 = 9 - 15 + 6 = 0 ✓. This verification confirms the accuracy of your solution.

What is a repeated root?

A repeated root (also called a double root) occurs when the discriminant equals zero, giving only one unique solution. This happens when the parabola just touches the x-axis at its vertex. Algebraically, both roots from the formula x = (-b ± √0) / 2a simplify to the same value: x = -b / 2a.

Can I use this for equations not in standard form?

First, rearrange your equation into standard form ax² + bx + c = 0. For example, if you have 3x² = 5x - 2, rearrange to 3x² - 5x + 2 = 0, giving a = 3, b = -5, c = 2. Then use the calculator with these coefficients. Always ensure all terms are on the left side and the equation equals zero.

What's the difference between exact and decimal forms?

The exact form uses fractions and radicals (like 3/2 or 1 + √5) and is mathematically precise. The decimal form (like 1.5 or 3.236068) is an approximation useful for practical applications. Exact forms are preferred in pure mathematics, while decimals are often used in engineering and applied sciences.

Can quadratic equations have more than two solutions?

No, by the Fundamental Theorem of Algebra, a polynomial equation of degree n has exactly n roots (counting multiplicities). Since quadratic equations are degree 2, they always have exactly two roots total. These may be two distinct real roots, one repeated real root (counted twice), or two complex conjugate roots, but never more than two.

Common Quadratic Formula Mistakes to Avoid

✗Forgetting the negative sign: Remember it's -b in the formula, not just b. If b = -5, then -b = 5.
✗Incorrect order of operations: Calculate b² - 4ac completely before taking the square root, not √b² - 4ac.
✗Dividing only part by 2a: The entire numerator (-b ± √discriminant) must be divided by 2a, not just part of it.
✗Missing the ± symbol: Don't forget to calculate both roots using + and - with the square root term.
✗Simplifying too early: Keep exact radical forms until the final answer; don't round intermediate steps.
✗Misidentifying coefficients: In 5 - 3x + 2x², rearrange first: 2x² - 3x + 5, so a = 2, b = -3, c = 5.
✗Ignoring negative discriminants: When b² - 4ac < 0, don't stop! Express roots as complex numbers with i.

Alternative Methods for Solving Quadratic Equations

Factoring Method

Express the quadratic as a product of two binomials: (mx + n)(px + q) = 0. Then use the zero product property: if AB = 0, then A = 0 or B = 0.

Example: x² - 5x + 6 = (x - 2)(x - 3) = 0
Therefore: x = 2 or x = 3

Best for: Simple integer coefficients that factor easily

Completing the Square

Rewrite the equation as a perfect square trinomial plus a constant: (x + p)² = q, then take square roots of both sides.

Example: x² + 6x + 5 = 0
→ (x + 3)² - 9 + 5 = 0
→ (x + 3)² = 4
→ x = -3 ± 2

Best for: Deriving the quadratic formula and theoretical understanding

Graphing Method

Plot the parabola y = ax² + bx + c and find where it intersects the x-axis (where y = 0). The x-coordinates of intersection points are the roots.

Example: Graph y = x² - 4x + 3
X-intercepts at (1, 0) and (3, 0)
Therefore: x = 1 or x = 3

Best for: Visual learners and approximate solutions