Solving Quadratic Equations: When the Discriminant is Negative

When faced with a quadratic equation, such as x2 2x 5 0, and the discriminant is negative, the equation may seem unsolvable using real numbers alone. However, by understanding the concept of complex numbers, we can find the solutions. This article will guide you through the process of solving such equations and the application of complex numbers.

Understanding the Problem

The given equation is x2 2x 5 0. Before we proceed with solving it, it's important to review how to determine the discriminant and what it means for the nature of the solutions.

Step 1: Identify the coefficients
In the equation x2 2x 5 0, the coefficients are:

a 1 b 2 c 5

Step 2: Calculate the discriminant
Using the formula D b2 - 4ac, we can determine the discriminant.

D 22 - 4(1)(5) 4 - 20 -16
Since the discriminant is negative (D -16), the solutions will be complex numbers.

Solving the Equation

There are two methods to solve quadratic equations with a negative discriminant: completing the square or using the quadratic formula.

Method 1: Completing the Square

Note: Before starting, subtract 5 from both sides to set the equation to 0: x2 2x -5

Step 1: Add 1 to both sides to complete the square

x2 2x 1 -4

Step 2: Factor the left side

(x 1)2 -4

Step 3: Take the square root of both sides

x 1 ±√(-4)

Step 4: Simplify the square root

x 1 ±2i

Step 5: Solve for x

x -1 ± 2i

Therefore, the solutions are x -1 2i and x -1 - 2i.

Method 2: Using the Quadratic Formula

The quadratic formula is given by:

x [-b ± √(b2 - 4ac)] / 2a

Substitute the values a 1, b 2, and c 5

x [-2 ± √(22 - 4(1)(5))] / 2(1)

x [-2 ± √(-16)] / 2

x [-2 ± 4i] / 2

x -1 ± 2i

Both methods yield the same solutions: x -1 2i and x -1 - 2i.

Verification

Let's verify the solutions by substituting them back into the original equation.

For x -1 2i

Step 1: Calculate x2

(-1 2i)2 1 - 4i 4i2

1 - 4i - 4 -3 - 4i

Step 2: Calculate 2x

2(-1 2i) -2 4i

Step 3: Combine all terms

(-3 - 4i) (-2 4i) 5 0

-3 - 2 5 - 4i 4i 0

0 0

Therefore, x -1 2i is a valid solution.

For x -1 - 2i

Step 1: Calculate x2

(-1 - 2i)2 1 4i 4i2

1 4i - 4 -3 4i

Step 2: Calculate 2x

2(-1 - 2i) -2 - 4i

Step 3: Combine all terms

(-3 4i) (-2 - 4i) 5 0

-3 - 2 5 4i - 4i 0

0 0

Therefore, x -1 - 2i is also a valid solution.

Conclusion

When the discriminant of a quadratic equation is negative, the solutions are complex numbers. By completing the square or using the quadratic formula, we can find the solutions and verify them. Understanding the use of complex numbers is crucial in solving such equations.

Key Concepts

Quadratic Equation: An equation of the form ax2 bx c 0 where a, b, c are constants and a ≠ 0. Discriminant: The value D b2 - 4ac which determines the nature of the solutions (real and distinct, real and equal, or complex). Complex Numbers: Numbers of the form a bi where a and b are real numbers, and i is the imaginary unit with i2 -1.

Keywords

quadratic equation, complex numbers, discriminant