Find All Negative Solutions Calculator

This calculator helps you find all negative solutions to quadratic equations. Whether you're solving for x when the quadratic equation has negative roots, this tool provides a clear and accurate solution.

How to Use This Calculator

To find all negative solutions to a quadratic equation, follow these simple steps:

Enter the coefficients of your quadratic equation in the form ax² + bx + c = 0.
Click the "Calculate" button to find all negative solutions.
Review the results and any additional information provided.

The calculator will display all negative solutions to the quadratic equation you entered. If there are no negative solutions, it will indicate that.

Formula Explained

The solutions to the quadratic equation ax² + bx + c = 0 are given by the quadratic formula:

x = [-b ± √(b² - 4ac)] / (2a)

To find all negative solutions, we need to find values of x that satisfy both the quadratic equation and x < 0.

The discriminant (D) of the quadratic equation is calculated as:

D = b² - 4ac

If D > 0, there are two distinct real roots. If D = 0, there is one real root (a repeated root). If D < 0, there are no real roots.

Worked Examples

Example 1: Two Negative Solutions

Consider the quadratic equation x² + 5x + 6 = 0.

Using the quadratic formula:

x = [-5 ± √(25 - 24)] / 2 x = [-5 ± √1] / 2 x = [-5 ± 1] / 2

This gives two solutions: x = -2 and x = -3. Both are negative, so they are the negative solutions.

Example 2: One Negative Solution

Consider the quadratic equation 2x² + 3x - 2 = 0.

Using the quadratic formula:

x = [-3 ± √(9 + 16)] / 4 x = [-3 ± √25] / 4 x = [-3 ± 5] / 4

This gives two solutions: x = 0.5 and x = -2. Only x = -2 is negative, so it is the negative solution.

Example 3: No Negative Solutions

Consider the quadratic equation x² - 3x + 2 = 0.

Using the quadratic formula:

x = [3 ± √(9 - 8)] / 2 x = [3 ± √1] / 2 x = [3 ± 1] / 2

This gives two solutions: x = 2 and x = 1. Both are positive, so there are no negative solutions.

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

How do I find the solutions to a quadratic equation?

You can find the solutions to a quadratic equation using the quadratic formula: x = [-b ± √(b² - 4ac)] / (2a).

What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation is the part of the quadratic formula under the square root: D = b² - 4ac. It determines the nature of the roots.

Can a quadratic equation have more than two solutions?

No, a quadratic equation can have at most two solutions, which can be real or complex. If the discriminant is positive, there are two distinct real solutions. If the discriminant is zero, there is one real solution. If the discriminant is negative, there are two complex solutions.