Solve Quadratics with Square Roots Calculator

How to Use This Calculator

Enter the coefficients of your quadratic equation in the form ax² + bx + c = 0. The calculator will solve for x using the quadratic formula:

The calculator handles square roots in the coefficients by treating them as mathematical expressions. Simply enter the square roots as √ followed by the radicand (the number inside the square root).

For example, to enter √2, type "√2". The calculator will evaluate the square roots before applying the quadratic formula.

Quadratic Formula with Square Roots

The quadratic formula is the standard method for solving quadratic equations. When coefficients include square roots, the formula remains the same:

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

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root.
• If the discriminant is negative, there are two complex roots.

The calculator automatically evaluates the discriminant and provides the appropriate roots.

Step-by-Step Solution

1. Enter the coefficients a, b, and c in the calculator. These can include square roots.
2. The calculator will evaluate any square roots in the coefficients.
3. Calculate the discriminant: b² - 4ac.
4. Take the square root of the discriminant.
5. Apply the quadratic formula to find the two roots.
6. The calculator will display the roots and any relevant information about the discriminant.

Worked Example

Let's solve the equation x² + √2x + √3 = 0.

1. Identify the coefficients: a = 1, b = √2, c = √3.
2. Calculate the discriminant: (√2)² - 4(1)(√3) = 2 - 4√3.
3. Since 2 - 4√3 is negative (√3 ≈ 1.732), the discriminant is negative.
4. Apply the quadratic formula:
   x = [-√2 ± √(2 - 4√3)] / 2
5. The roots are complex numbers:
   x = -√2/2 ± i√(2√3 - 1)/2

This example shows how the calculator handles square roots in the coefficients and provides complex roots when the discriminant is negative.

Interpreting Results

The calculator provides the roots of the quadratic equation. The interpretation depends on the nature of the roots:

• Real roots: The equation crosses the x-axis at these points. For example, in x² - 5x + 6 = 0, the roots are x=2 and x=3.
• Complex roots: The equation does not cross the x-axis. The roots are complex numbers. For example, in x² + 1 = 0, the roots are x=i and x=-i.

The calculator also provides the discriminant value, which indicates the nature of the roots.