Quadratic Equations with Complex Solutions

When solving quadratic equations, you might encounter situations where the quadratic formula requires taking the square root of a negative number. This is where complex numbers come into play.

The Discriminant and Imaginary Numbers

For any quadratic equation in the form $ax^2 + bx + c = 0$, the roots are found using the quadratic formula:

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

The expression inside the square root, $b^2 - 4ac$, is called the discriminant.

• If $b^2 - 4ac > 0$, there are two real roots.
• If $b^2 - 4ac = 0$, there is one real root.
• If $b^2 - 4ac < 0$, there are two complex roots.

To evaluate the square root of a negative number, we use the imaginary unit $i$, where $i^2 = -1$ (meaning $i = \sqrt{-1}$).

Complex Conjugate Pairs

An important rule in algebra is that if a polynomial has entirely real coefficients, its complex roots will always come in conjugate pairs. This means if $a + bi$ is a root, then $a - bi$ must also be a root.

Example 1: Solving a Quadratic Equation

Problem: Solve $x^2 + 4x + 13 = 0$.

Solution: Here, $a = 1$, $b = 4$, and $c = 13$.

First, find the discriminant: $b^2 - 4ac = 4^2 - 4(1)(13) = 16 - 52 = -36$

Now, plug this into the quadratic formula: $x = \frac{-4 \pm \sqrt{-36}}{2}$

Since $\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i$, we have: $x = \frac{-4 \pm 6i}{2}$

Divide both terms in the numerator by 2: $x = -2 \pm 3i$

The solutions are the complex conjugate pair $-2 + 3i$ and $-2 - 3i$.

Example 2: Finding an Equation from Complex Roots

Problem: Find a quadratic equation with roots $2 + 3i$ and $2 - 3i$.

Solution: If $r_1$ and $r_2$ are roots of a quadratic equation, the equation can be written as: $x^2 - (r_1 + r_2)x + (r_1 \cdot r_2) = 0$

1. Find the sum of the roots: $(2 + 3i) + (2 - 3i) = 4$

2. Find the product of the roots: $(2 + 3i)(2 - 3i) = 2^2 - (3i)^2 = 4 - 9i^2$ Since $i^2 = -1$, the product is: $4 - 9(-1) = 4 + 9 = 13$

3. Write the equation: Substitute the sum and product back into the formula: $x^2 - 4x + 13 = 0$