Question

1. janie was asked to multiply the complex numbers. find the error and correctly solve.

(3 + 2i)(5 - i) = 15 - 3i + 10i - 2i²
= 15 + 7i - 2i²
= -2i² + 7i + 15
error:
correct answer:

1. solve the following completely: $x^2 + 25 = 0$

Explanation:

Problem 14

Step1: Recall \(i^2 = -1\)

Janie's error was not substituting \(i^2 = -1\) at the end. She stopped at \( -2i^2 + 7i + 15 \) without simplifying \(i^2\).

Step2: Substitute \(i^2 = -1\)

Start from her last correct step: \( -2i^2 + 7i + 15 \). Substitute \(i^2 = -1\):
\( -2(-1) + 7i + 15 = 2 + 7i + 15 \)

Step3: Combine like terms

Combine the real numbers: \( 2 + 15 + 7i = 17 + 7i \)

Error: Janie did not substitute \(i^2 = -1\) to simplify the expression.

Correct answer: \(17 + 7i\)

Problem 16

Step1: Isolate \(x^2\)

Start with the equation \(x^2 + 25 = 0\). Subtract 25 from both sides:
\(x^2 = -25\)

Step2: Take square root (use \(i\))

Take the square root of both sides. Recall that \(\sqrt{-a} = i\sqrt{a}\) for \(a > 0\). So:
\(x = \pm\sqrt{-25} = \pm i\sqrt{25} = \pm 5i\)

Answer:

\(x = \pm 5i\)