Quadratic Equation Calculator

Published on: December 8, 2024

Final Answer: Free • Full Steps: Plus

This Quadratic Equation Calculator helps you solve equations of the form ax² + bx + c = 0 step by step. First write the equation in standard form, then solve it using a suitable method such as factoring, completing the square, or the quadratic formula. Finally check the roots in the original equation if needed. It is a simple way to check answers, understand quadratic solving, and practise basic algebra step by step.

Step-by-step method

Rewrite the equation in standard form ax^2 + bx + c = 0.
Identify the coefficients a, b, and c.
Compute the discriminant D = b^2 − 4ac.
Use the quadratic formula x = (−b ± √D) / (2a).
Simplify the two solutions.

Example 1:

\(4x^{2}-5x-12=0\)

Step 1 - Rewrite the equation in standard form ax^2 + bx + c = 0.

In this problem: Rewrite \(4x^{2}-5x-12=0\) into standard form.

\(4 x^{2} - 5 x - 12 = 0\)

Step 2 - Identify the coefficients a, b, and c.

In this problem: Read off the coefficients from \(ax^2 + bx + c = 0\).

\(a = 4, \quad b = -5, \quad c = -12\)

Step 3 - Compute the discriminant D = b^2 − 4ac.

In this problem: Compute the discriminant.

\(D = \left(-5\right)^{2} - 4 \cdot 4 \cdot \left(-12\right) = 217\)

Step 4 - Use the quadratic formula x = (−b ± √D) / (2a).

In this problem: Use the quadratic formula with the values found.

\(x = \frac{5 \pm \sqrt{217}}{8}\)

Step 5 - Simplify the two solutions.

In this problem: Simplify the solutions.

\(x = \frac{5}{8} + \frac{\sqrt{217}}{8} \quad\text{or}\quad x = \frac{5}{8} - \frac{\sqrt{217}}{8}\)

Final answer:

\(x = \frac{5}{8} + \frac{\sqrt{217}}{8} \quad\text{or}\quad x = \frac{5}{8} - \frac{\sqrt{217}}{8}\)

Example 2:

\(x^{2}+6x+5=0\)

Step 1 - Rewrite the equation in standard form ax^2 + bx + c = 0.

In this problem: Rewrite \(x^{2}+6x+5=0\) into standard form.

\(x^{2} + 6 x + 5 = 0\)