Solving Quadratic by Square Roots Calculator

Quadratic equations are fundamental in algebra, and solving them by square roots is a common method. This calculator helps you solve quadratic equations of the form ax² + bx + c = 0 using the square root method when the equation can be factored.

How to Solve Quadratic Equations by Square Roots

The square root method for solving quadratic equations is applicable when the equation can be written in the form (x + a)² = b. Here's how to solve it:

Start with the quadratic equation in standard form: ax² + bx + c = 0.
Move the constant term to the other side: ax² + bx = -c.
Divide all terms by the coefficient of x² (a) to make the coefficient of x² equal to 1: x² + (b/a)x = -c/a.
Add (b/2a)² to both sides to complete the square: x² + (b/a)x + (b/2a)² = -c/a + (b/2a)².
Write the left side as a perfect square: (x + b/2a)² = -c/a + (b²/4a²).
Take the square root of both sides: x + b/2a = ±√(-c/a + b²/4a²).
Simplify the expression under the square root: x + b/2a = ±√(b² - 4ac)/2a.
Solve for x: x = [-b ± √(b² - 4ac)] / 2a.

This method works best when the quadratic can be easily factored or when the discriminant (b² - 4ac) is a perfect square.

The Quadratic Formula

The general solution to the quadratic equation ax² + bx + c = 0 is given by:

x = [-b ± √(b² - 4ac)] / 2a

Where:

a, b, and c are coefficients of the quadratic equation
√(b² - 4ac) is the discriminant
The ± symbol indicates two possible solutions

The discriminant determines the nature of the roots:

If b² - 4ac > 0: Two distinct real roots
If b² - 4ac = 0: One real root (repeated)
If b² - 4ac < 0: Two complex conjugate roots

Worked Example

Let's solve the quadratic equation x² - 5x + 6 = 0 using the square root method.

Start with the equation: x² - 5x + 6 = 0
Move the constant term: x² - 5x = -6
Divide by coefficient of x²: x² - 5x = -6
Add (b/2a)² = (5/2)² = 6.25 to both sides: x² - 5x + 6.25 = -6 + 6.25
Write as perfect square: (x - 2.5)² = 0.25
Take square roots: x - 2.5 = ±0.5
Solve for x: x = 2.5 ± 0.5
Final solutions: x = 3 and x = 2

This example shows how the square root method can be applied to find the roots of a quadratic equation.

Frequently Asked Questions

When should I use the square root method to solve quadratic equations?

The square root method is most effective when the quadratic equation can be easily factored or when the discriminant is a perfect square. It's particularly useful when the equation can be written in the form (x + a)² = b.

What is the discriminant in the quadratic formula?

The discriminant is the part of the quadratic formula under the square root: b² - 4ac. It determines the nature of the roots: real and distinct, real and equal, or complex.

Can the square root method solve all quadratic equations?

No, the square root method is most effective when the equation can be easily factored or when the discriminant is a perfect square. For more complex equations, other methods like completing the square or using the quadratic formula may be more appropriate.

What if the discriminant is negative?

If the discriminant is negative, the equation has two complex conjugate roots. These can still be found using the quadratic formula, but they involve imaginary numbers.