Advanced Quadratic Equation Solver with Graph | Mathematical Tools

Thinkforu.org Quadratic Equation Solver

Enter Equation Coefficients

For equation: ax² + bx + c = 0

Coefficient a:

Coefficient b:

Coefficient c:

Solution

Enter coefficients and click solve to see the solution.

Complete Guide to Solving Quadratic Equations

The Quadratic Formula

x = [-b ± √(b² - 4ac)] / (2a)

Example 1: Simple Quadratic Equation

Let's solve: x² + 5x + 6 = 0

Here, a = 1, b = 5, c = 6

Using the quadratic formula:

x = [-5 ± √(5² - 4(1)(6))] / (2(1))

x = [-5 ± √(25 - 24)] / 2

x = [-5 ± √1] / 2

x = [-5 ± 1] / 2

x = -2 or -3

Example 2: Complex Roots

Let's solve: x² + 2x + 5 = 0

Here, a = 1, b = 2, c = 5

Using the quadratic formula:

x = [-2 ± √(4 - 20)] / 2

x = [-2 ± √(-16)] / 2

x = -1 ± 2i

Pro Tips for Solving Quadratic Equations

💡

Factoring First:

Before using the quadratic formula, check if the equation can be factored easily.

📊

Understanding the Graph:

The roots are where the parabola crosses the x-axis. The vertex tells you the minimum or maximum point.

🎯

Discriminant Check:

b² - 4ac determines the type of roots:
• > 0: Two real roots
• = 0: One real root
• < 0: Two complex roots

Step-by-Step Guide

1. Identify the Coefficients

Write your equation in standard form (ax² + bx + c = 0) and identify a, b, and c.

2. Check the Discriminant

Calculate b² - 4ac to determine the type of roots you'll get.

3. Apply the Formula

Use the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a)

4. Simplify the Result

Reduce fractions and simplify radicals if possible.

5. Verify Your Answer

Plug your solutions back into the original equation to check.

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2, written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Why can't 'a' be zero in a quadratic equation?

If 'a' were zero, the equation would become bx + c = 0, which is a linear equation, not a quadratic equation. The term ax² is what makes the equation quadratic.

What does the discriminant tell us?

The discriminant (b² - 4ac) tells us about the nature of the roots:
• If positive: Two distinct real roots
• If zero: One repeated real root
• If negative: Two complex conjugate roots

What is the vertex of a parabola?

The vertex is the highest or lowest point of the parabola. For a quadratic function f(x) = ax² + bx + c, the x-coordinate of the vertex is x = -b/(2a), and it represents either a maximum (if a < 0) or minimum (if a > 0) point of the function.

How do I know if my answers are correct?

You can verify your answers by substituting them back into the original equation. If the equation equals zero when you plug in your solution, the answer is correct. Our calculator automatically verifies the solutions for you.

What are complex roots?

Complex roots occur when the discriminant is negative, meaning the parabola never crosses the x-axis. These roots contain i, the imaginary unit, where i² = -1. They always come in conjugate pairs (a + bi and a - bi).

Why does the graph show a parabola?

The graph of a quadratic equation is always a parabola - a U-shaped curve that opens upward if a > 0 or downward if a < 0. The points where the parabola crosses the x-axis are the roots of the equation.

Quadratic Equation Solver