Quadratic Equations with Imaginary Solutions

Quadratic Equations with Imaginary Solutions

Quadratic equations are equations of the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations can have real or complex solutions. Complex solutions are solutions that involve the imaginary unit, i, where i² = -1. In this article, we will explore how to solve quadratic equations with imaginary solutions using the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is derived from completing the square and can be used to find the solutions of any quadratic equation. The formula is as follows:

x = (-b ± √(b² - 4ac)) / 2a

Where:

• a, b, and c are the coefficients of the quadratic equation.
• x is the solution to the equation.
• ± indicates that there are two possible solutions, one with a plus sign and one with a minus sign.

Solving Quadratic Equations with Imaginary Solutions

When the discriminant (b² - 4ac) in the quadratic formula is negative, the quadratic equation has imaginary solutions. This is because the square root of a negative number is imaginary.

Here's how to solve quadratic equations with imaginary solutions:

1. Identify the coefficients a, b, and c of the quadratic equation.
2. Calculate the discriminant (b² - 4ac). If the discriminant is negative, the equation has imaginary solutions.
3. Substitute the values of a, b, and c into the quadratic formula.
4. Simplify the expression under the square root by factoring out -1 and taking the square root of -1, which is i.
5. Write the solutions in the form a + bi, where a and b are real numbers and i is the imaginary unit.

Example

Let's solve the quadratic equation x² + 2x + 5 = 0.

1. Identify the coefficients: a = 1, b = 2, and c = 5.
2. Calculate the discriminant: b² - 4ac = 2² - 4(1)(5) = -16. Since the discriminant is negative, the equation has imaginary solutions.
3. Substitute the values into the quadratic formula: x = (-2 ± √(-16)) / 2(1).
4. Simplify the expression under the square root: x = (-2 ± √(-1) * √16) / 2 = (-2 ± 4i) / 2.
5. Write the solutions: x = -1 + 2i and x = -1 - 2i.

Conclusion

Solving quadratic equations with imaginary solutions using the quadratic formula is a straightforward process. By following the steps outlined above, you can successfully find the complex solutions to any quadratic equation.