Problem 5 - Precalculus Student Solutions Manual 5th

Question

Match the equation in Column I with its solution \((s)\) in Column II. A. \(\pm 5 i\) B. \(\pm 2 \sqrt{5}\) C. \(\pm i \sqrt{5}\) D. \(5\) E. \(\pm \sqrt{5} \quad\) F. \(-5\) G. \(\pm 5\) H. \(\pm 2 i \sqrt{5}\) $$x^{2}=-20$$

Step-by-Step Solution

Answer

H. \( \pm 2 i \sqrt{5} \)

Step 1 - Understand the Given Equation

The equation given is \( x^2 = -20 \). This is a quadratic equation.

Step 2 - Identify Nature of Roots

Since the right-hand side of the equation is negative (\( -20 \)), the solutions will be complex numbers containing \'i\' (imaginary unit).

Step 3 - Solve for x

To solve for \( x \), take the square root on both sides of the equation: \[ x^2 = -20 \ x = \pm \sqrt{-20} \]

Step 4 - Simplify the Square Root

Since \( -20 \) can be written as \( -1 \times 20 \), we can separate the square root: \[ \sqrt{-20} = \sqrt{-1 \times 20} = \sqrt{-1} \times \sqrt{20} = i \times \sqrt{20} \]

Step 5 - Write the Final Solution

Recognize that \( \sqrt{20} \) can be simplified to \( 2 \sqrt{5} \). Thus: \[ x = \pm i \times 2 \sqrt{5} \]

Step 6 - Match With Column II

The simplified solution matches option H: \( \pm 2 i \sqrt{5} \).

Key Concepts

Complex Numbers

Complex numbers extend the idea of one-dimensional numbers like integers and real numbers to two dimensions. A complex number is typically written in the form \(a + bi\), where:

• \