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Solving Quadratic Equations by Finding Square Roots Calculator

Reviewed by Calculator Editorial Team

A quadratic equation is a second-degree polynomial equation in a single variable. The general form is ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. Solving quadratic equations is essential in algebra, physics, engineering, and many other fields.

What is a Quadratic Equation?

Quadratic equations are polynomial equations of degree 2. They have the general form:

ax² + bx + c = 0

where:

  • a, b, c are constants
  • a ≠ 0 (if a = 0, it becomes a linear equation)
  • x is the variable

Quadratic equations can be solved by several methods, including:

  • Factoring
  • Completing the square
  • Using the quadratic formula
  • Graphical methods

This guide focuses on solving quadratic equations by finding square roots, which is equivalent to using the quadratic formula.

Solving by Finding Square Roots

The most common method for solving quadratic equations is using the quadratic formula, which involves finding the square roots of a number. This method works for all quadratic equations and provides exact solutions when the discriminant is a perfect square.

The quadratic formula is derived from completing the square and is expressed as:

x = [-b ± √(b² - 4ac)] / (2a)

Where:

  • x is the variable
  • a, b, c are coefficients from the quadratic equation
  • √(b² - 4ac) is the square root of the discriminant

The discriminant (b² - 4ac) determines the nature of the roots:

  • If discriminant > 0: Two distinct real roots
  • If discriminant = 0: One real root (repeated)
  • If discriminant < 0: Two complex conjugate roots

The Quadratic Formula

The quadratic formula is the most reliable method for solving quadratic equations. It works for all quadratic equations and provides exact solutions when the discriminant is a perfect square.

x = [-b ± √(b² - 4ac)] / (2a)

To use the quadratic formula:

  1. Identify the coefficients a, b, and c from the quadratic equation
  2. Calculate the discriminant (b² - 4ac)
  3. Find the square root of the discriminant
  4. Apply the quadratic formula to find the two solutions

The ± symbol indicates that there are two solutions, one with the positive square root and one with the negative square root.

Worked Example

Let's solve the quadratic equation 2x² + 5x - 3 = 0 using the quadratic formula.

Given: 2x² + 5x - 3 = 0

a = 2, b = 5, c = -3

Step 1: Calculate the discriminant

Discriminant = b² - 4ac = (5)² - 4(2)(-3) = 25 + 24 = 49

Step 2: Find the square roots

√49 = ±7

Step 3: Apply the quadratic formula

x = [-b ± √(b² - 4ac)] / (2a) = [-5 ± 7] / 4

This gives two solutions:

  • x = (-5 + 7)/4 = 2/4 = 0.5
  • x = (-5 - 7)/4 = -12/4 = -3

Therefore, the solutions to the equation 2x² + 5x - 3 = 0 are x = 0.5 and x = -3.

Frequently Asked Questions

What is the quadratic formula?

The quadratic formula is x = [-b ± √(b² - 4ac)] / (2a), where a, b, and c are coefficients from the quadratic equation ax² + bx + c = 0.

When should I use the quadratic formula?

You should use the quadratic formula when the quadratic equation cannot be easily factored or when you need exact solutions, especially when the discriminant is a perfect square.

What does the discriminant tell me?

The discriminant (b² - 4ac) tells you about the nature of the roots of the quadratic equation:

  • Positive discriminant: Two distinct real roots
  • Zero discriminant: One real root (repeated)
  • Negative discriminant: Two complex conjugate roots