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Chapter 8: Problem 13

Find all complex-number solutions. $$ 9 x^{2}-49=0 $$

Short Answer

Expert verified
The solutions are \(x = \frac{7}{3}\) and \(x = -\frac{7}{3}\).

Step by step solution

01

- Move constant to the other side

First, we need to isolate the term with the variable. Move the constant 49 to the other side of the equation by adding 49 to both sides: \[9x^2 = 49\]
02

- Divide by the coefficient of x^2

Next, divide both sides by the coefficient of \(x^2\), which is 9, to simplify the equation: \[x^2 = \frac{49}{9}\]
03

- Take the square root of both sides

To solve for \(x\), take the square root of both sides of the equation. Remember to consider both the positive and negative roots: \[x = \pm \sqrt{\frac{49}{9}}\]
04

- Simplify the square root

Simplify the square root of the fraction by taking the square root of the numerator and the denominator separately: \[x = \pm \frac{\sqrt{49}}{\sqrt{9}} = \pm \frac{7}{3}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. They are written in the form of \(a + bi\), where:
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