Quadratics with Complex Solutions - Algebra-1

1. Fundamental Concepts

Definition: Quadratic equations are polynomial equations of the form $$ax^2 + bx + c = 0$$ , where $$a \neq 0$$ .
Complex Solutions: When the discriminant ( $$b^2 - 4ac$$ ) is negative, the solutions to a quadratic equation are complex numbers.
Imaginary Unit: The imaginary unit $$i$$ is defined as $$i = \sqrt{-1}$$ .

2. Key Concepts

Quadratic Formula: $$x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}$$

Discriminant: The discriminant $$b^2 - 4ac$$ determines the nature of the roots: real and distinct, real and equal, or complex.

Solving with Complex Numbers: When $$b^2 - 4ac < 0$$ , the solutions involve the imaginary unit $$i$$ .

3. Examples

Example 1 (Basic)

Problem: Solve the quadratic equation $$x^2 + 4x + 5 = 0$$ .

Step-by-Step Solution:

Identify coefficients: $$a = 1$$ , $$b = 4$$ , $$c = 5$$ .
Calculate the discriminant: $$b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 5 = 16 - 20 = -4$$ .
Apply the quadratic formula: $$x = \frac{{-4 \pm \sqrt{{-4}}}}{{2 \cdot 1}} = \frac{{-4 \pm 2i}}{{2}} = -2 \pm i$$ .

Validation: Substitute $$x = -2 + i$$ into the original equation: $$( -2 + i )^2 + 4( -2 + i ) + 5 = (-2)^2 + 2(-2)(i) + (i)^2 + 4(-2) + 4(i) + 5 = 4 - 4i - 1 - 8 + 4i + 5 = 0$$ ✓

Example 2 (Intermediate)

Problem: Solve the quadratic equation $$2x^2 - 4x + 5 = 0$$ .

Step-by-Step Solution:

Identify coefficients: $$a = 2$$ , $$b = -4$$ , $$c = 5$$ .
Calculate the discriminant: $$b^2 - 4ac = (-4)^2 - 4 \cdot 2 \cdot 5 = 16 - 40 = -24$$ .
Apply the quadratic formula: $$x = \frac{{4 \pm \sqrt{{-24}}}}{{4}} = \frac{{4 \pm 2\sqrt{6}i}}{{4}} = 1 \pm \frac{\sqrt{6}}{2}i$$ .

Validation: Substitute $$x = 1 + \frac{\sqrt{6}}{2}i$$ into the original equation: $$2(1 + \frac{\sqrt{6}}{2}i)^2 - 4(1 + \frac{\sqrt{6}}{2}i) + 5 = 2(1 + \sqrt{6}i + \frac{3}{2}i^2) - 4 - 2\sqrt{6}i + 5 = 2(1 + \sqrt{6}i - \frac{3}{2}) - 4 - 2\sqrt{6}i + 5 = 0$$ ✓

4. Problem-Solving Techniques

Check Discriminant First: Always calculate the discriminant to determine the type of solutions before applying the quadratic formula.
Use Imaginary Unit Carefully: Remember that $$i^2 = -1$$ when simplifying expressions involving $$i$$ .
Substitute Back: After finding the solutions, substitute them back into the original equation to verify correctness.