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Chapter 8: Problem 27

Find all complex-number solutions. $$ \left(y+\frac{3}{4}\right)^{2}=\frac{17}{16} $$

Short Answer

Expert verified
y = \frac{\sqrt{17} - 3}{4} \, \frac{-\sqrt{17} - 3}{4}

Step by step solution

01

Isolate the squared term

Start by taking the square root of both sides to isolate the term inside the parentheses. This gives us: \[\text{Taking square root of both sides:} \rightarrow \sqrt{(y + \frac{3}{4})^2} = \sqrt{\frac{17}{16}} \rightarrow y + \frac{3}{4} = \pm\frac{\sqrt{17}}{4}\]
02

Solve for y by creating two equations

Since the square root can have positive and negative values, create two separate linear equations:1. \[ y + \frac{3}{4} = \frac{\sqrt{17}}{4} \text{This implies:} \y = \frac{\sqrt{17}}{4} - \frac{3}{4} \y = \frac{\sqrt{17} - 3}{4} \]2. \[ y + \frac{3}{4} = -\frac{\sqrt{17}}{4} \text{This implies:} \y = -\frac{\sqrt{17}}{4} - \frac{3}{4} \y = \frac{-\sqrt{17} - 3}{4} \]
03

Write the final solutions

The two solutions for the equation are: \[ y = \frac{\sqrt{17} - 3}{4} \ y = \frac{-\sqrt{17} - 3}{4} \]

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Key Concepts

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

Solving Equations
When solving equations, the goal is to find the value of the variable that satisfies the equation. In this problem, the equation is \ \ \( \left(y+\frac{3}{4}\right)^{2}=\frac{17}{16} \). To solve it, we need to isolate the variable \ \( y \). Let's break it down step by step. First, we isolate the squared term by taking the square root of both sides of the equation. This step helps us get rid of the exponent and simplifies the process.\
Square Root Property
The square root property is a powerful tool for solving quadratic equations of the form \ \( (ax + b)^2 = c \). When we take the square root of both sides, we must remember to account for both the positive and negative square roots. In this exercise, we took the square root of both sides: \ \ \( \sqrt{(y + \frac{3}{4})^2} = \sqrt{\frac{17}{16}} \rightarrow y + \frac{3}{4} = \pm\frac{\sqrt{17}}{4} \). This introduces two separate equations: one with the positive root and one with the negative root.\ \ We then solve each equation separately to find the possible values of \ \( y \).
Linear Equations
Linear equations are the simplest form of equations and involve variables raised only to the first power. After using the square root property, we get two linear equations to solve: \ \ \( y + \frac{3}{4} = \frac{\sqrt{17}}{4} \rightarrow y = \frac{\sqrt{17} - 3}{4} \)
and
\( y + \frac{3}{4} = -\frac{\sqrt{17}}{4} \rightarrow y = \frac{-\sqrt{17} - 3}{4} \). \ \ Solving these linear equations is straightforward. We isolate \ \( y \) by moving the constant terms to the other side. This gives us two solutions, which are the final complex-number solutions for the original equation.

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