Problem 41 - Precalculus: Functions and Graphs

Question

Find the solutions of the equation. $$x^{2}+4 x+13=0$$

Step-by-Step Solution

Answer

The solutions are complex: \(-2 + 3i\) and \(-2 - 3i\).

Step 1: Identify the coefficients

Given the quadratic equation is \(x^2 + 4x + 13 = 0\). Identify the coefficients: \(a = 1\), \(b = 4\), and \(c = 13\).

Step 2: Calculate the discriminant

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(\Delta = b^2 - 4ac\). Substitute the coefficients into this formula: \(\Delta = 4^2 - 4 \times 1 \times 13\). Calculate: \(\Delta = 16 - 52 = -36\).

Step 3: Determine the nature of the roots

Since the discriminant \(\Delta = -36\) is less than zero, the quadratic equation \(x^2 + 4x + 13 = 0\) has complex roots.

Step 4: Use the quadratic formula

The quadratic formula to find the roots is \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\). Substitute the values: \(x = \frac{-4 \pm \sqrt{-36}}{2}\).

Step 5: Solve for the complex roots

Calculate \(\sqrt{-36} = 6i\), where \(i\) is the imaginary unit. Substitute into the formula: \(x = \frac{-4 \pm 6i}{2}\). Simplify to find the roots: \(x = -2 \pm 3i\).

Key Concepts

Discriminant

The discriminant is a crucial component in determining the nature of the roots for any quadratic equation. It's found using the formula \(\Delta = b^2 - 4ac\), where \(a\), \(b\), and \(c\) are coefficients from \(ax^2 + bx + c = 0\). In this equation:

• \(a = 1\)
• \(b = 4\)
• \(c = 13\)

To calculate the discriminant, we substitute: \(\Delta = 4^2 - 4 \times 1 \times 13 = 16 - 52 = -36\).
Since \(\Delta = -36\) is negative, it tells us the quadratic equation will have complex roots instead of real ones.
The sign of the discriminant is an essential clue:

• Positive: Two distinct real roots
• Zero: One real root (repeated)
• Negative: No real roots – results in complex roots

Complex Numbers

When a quadratic equation has a negative discriminant, as in this case with \(\Delta = -36\), the solutions are complex numbers. Complex numbers take the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit satisfying \(i^2 = -1\).
For the given equation, the discriminant's square root is \(\sqrt{-36} = 6i\). This involves understanding that \(\sqrt{-1} = i\). Complex roots often come in conjugate pairs – if \(a + bi\) is a root, then \(a - bi\) is also a root.
Understanding complex numbers helps:

• Explaining phenomena that real numbers cannot
• Providing solutions to equations that thought to have no "real" answers

Quadratic Formula

The quadratic formula is an invaluable tool for solving any quadratic equation. It's expressed as \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\). Here, it's used to find the roots of \(x^2 + 4x + 13 = 0\).
By substituting the known values:

• \(b = 4\)
• \(a = 1\)
• \(\Delta = -36\)

We have:\[x = \frac{-4 \pm \sqrt{-36}}{2 \times 1} = \frac{-4 \pm 6i}{2}\]
Simplifying yields the two complex solutions \(-2 + 3i\) and \(-2 - 3i\). The quadratic formula not only helps in getting the precise solutions but also indicates the nature of the solutions based on the discriminant value.
Steps to remember:

• Plug coefficients into the formula
• Handle the square root of the discriminant
• Simplify the expression carefully