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Chapter 9: Problem 28

Solve each equation. $$\left(x+\frac{1}{5}\right)^{2}=\frac{7}{25}$$

Short Answer

Expert verified
The solutions are \( x = - \frac{1}{5} + \frac{\sqrt{7}}{5} \) and \( x = - \frac{1}{5} - \frac{\sqrt{7}}{5} \).

Step by step solution

01

Take the square root of both sides

To eliminate the square on the left side of the equation, take the square root of both sides: \(\sqrt{\left(x+\frac{1}{5}\right)^{2}} = \sqrt{\frac{7}{25}}\).
02

Simplify the equation

Simplify the square root on both sides: \( x + \frac{1}{5} = \pm \frac{\sqrt{7}}{5} \).
03

Solve for x

Isolate \(x\) by subtracting \(\frac{1}{5}\) from both sides of the equation: \( x = - \frac{1}{5} \pm \frac{\sqrt{7}}{5} \).
04

Write the solutions

Write the two possible solutions for the equation: \[ x = - \frac{1}{5} + \frac{\sqrt{7}}{5} \] and \[ x = - \frac{1}{5} - \frac{\sqrt{7}}{5} \].

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Key Concepts

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

square root method
Starting with \(\left(x+\frac{1}{5}\right)^{2}=\frac{7}{25}\), we first apply the square root method. The goal here is to undo the square and get closer to isolating the variable. To do this, take the square root of both sides of the equation: \(\sqrt{\left(x+\frac{1}{5}\right)^{2}} = \sqrt{\frac{7}{25}}\). Remember that when you take the square root of both sides, you need to consider both the positive and negative solutions. So, we get: \( x + \frac{1}{5} = \pm \frac{\sqrt{7}}{5}\).This is a crucial step because it simplifies the quadratic equation into a linear form that's easier to solve.
simplifying equations
Once you've taken the square root, you'll have something like \( x + \frac{1}{5} = \pm \frac{\sqrt{7}}{5}\). The equation is simplified, but we still need to isolate the variable. Always check if the numbers can be simplified further. In our case, \( \pm \frac{\sqrt{7}}{5}\) is already in its simplest form. Simplifying equations helps in reducing complexity and making the calculation steps clearer. This step ensures that we are working with the simplest form of the equation, making it easier to find the final solution.
isolating variables
To find the value of \(x\), we need to isolate it on one side of the equation. Starting from \( x + \frac{1}{5} = \pm \frac{\sqrt{7}}{5}\), subtract \(\frac{1}{5}\) from both sides to get \(x\) alone: \( x = \pm \frac{\sqrt{7}}{5} - \frac{1}{5} \). This isolates \(x\), making it the subject of the equation. Isolation is important because it directly leads us to the solution without any further unnecessary steps. Essentially, isolation simplifies the equation down to its most basic form, making it easier to solve.
writing solutions
Now that we have \( x = - \frac{1}{5} \pm \frac{\sqrt{7}}{5} \), we must write the two possible solutions explicitly. This is because \(\pm\) indicates that there are two scenarios to consider:
  • \( x = - \frac{1}{5} + \frac{\sqrt{7}}{5} \).
  • \( x = - \frac{1}{5} - \frac{\sqrt{7}}{5} \).
Writing out both solutions ensures clarity and confirms that you've accounted for both possible values of \(x\). This step is crucial as it validates the completeness of your solution and ensures no potential answer is missed.

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Most popular questions from this chapter

Rewrite each function in the \(f(x)=a(x-h)^{2}+k\) form by completing the square. $$f(x)=-3 x^{2}-12 x-5$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it using properties. $$f(x)=-x^{2}+2 x-4$$

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Graph each function using transformations. $$f(x)=(x-4)^{2}-3$$

Solve each inequality algebraically and write any solution in interval notation. $$-2 x^{2}+8 x-10<0$$
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