/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 1 Solve the given equation. $$\s... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 1

Solve the given equation. $$\sqrt{x}=5$$

Short Answer

Expert verified
\( x = 25 \)

Step by step solution

01

Understand the Equation

The given equation is \(\text{\textbackslash sqrt{x}}=5\). This means that the square root of \(x\) is equal to 5.
02

Square Both Sides

To solve for \(x\), we need to eliminate the square root. Do this by squaring both sides of the equation: \[ \text{\textbackslash (} \text{\textbackslash sqrt{x}} = 5 \text{\textbackslash )} \onumber \text{\text \textbackslash implies} \text{\textbackslash (} (\text{\textbackslash sqrt{x}})^2 = 5^2 \text{\textbackslash )} \onumber \text{\textbackslash (} x = 25 \text{\textbackslash )} \onumber \]
03

Verify the Solution

Verify the solution by substituting \(x = 25 \) back into the original equation. \(\text{\textbackslash sqrt{25}} = 5\), which is a true statement. Thus, \(x = 25\) is the correct solution.

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Key Concepts

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

square root
The concept of the square root is essential in algebra. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5, because \(5 \times 5 = 25\).

Square roots are typically represented using the symbol \( \text{\backslash sqrt{}} \). So, \(\sqrt{25} = 5\). Understanding this concept helps in solving many problems in algebra, as it allows us to work backward from a squared number to its base value.

In our exercise, we see the equation \(\sqrt{x}=5\). This tells us that the square root of \(x\) is equal to 5. To find \(x\), we need to remove the square root.
squaring both sides
To eliminate a square root and solve for the variable inside, we use a technique called 'squaring both sides' of the equation. This involves raising both sides of the equation to the power of 2.

In our exercise, we have the equation \(\sqrt{x}=5\). To eliminate the square root, we square both sides as follows: \[\begin{aligned} (\sqrt{x})^2 & = 5^2 \ x & = 25 \end{aligned}\]

This step removes the square root, allowing us to find the value of \(x\). It's important to do the squaring correctly on both sides to maintain the equality. For our problem, squaring both sides results in the equation \(x=25\), which is our potential solution.
verifying solutions in algebra
Verification is a crucial step in solving algebraic equations. After finding a potential solution, we substitute it back into the original equation to ensure it satisfies the equation.

For our exercise, we've determined that \(x = 25\). To verify, we substitute \(x = 25\) back into the original square root equation: \[\sqrt{25}=5\]

Since \(\sqrt{25} = 5\) is a true statement, our solution \(x=25\) is correct. This step not only ensures our solution is accurate but also builds confidence in our problem-solving abilities.

For every algebraic equation you solve, always take the time to verify your answers.

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Most popular questions from this chapter

Sketch the graph of each equation in a rectangular coordinate system. Label the intercepts. $$y+2=0$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$\sqrt{25}+\sqrt{24}$$

In each of the following, perform the indicated operations and simplify as completely as possible. Assume all variables appearing under radical signs are non negative. $$8 \sqrt{25}-3 \sqrt{21}$$

Perform the indicated operations. Simplify all answers as completely as possible. Assume that all variables appearing under radical signs are nonnegative. $$\frac{6}{\sqrt{x}+\sqrt{3}}$$

Estimate the given square root between two consecutive integers without using a calculator, then use a calculator to find the square root rounded to two decimal places to confirm your estimate. $$\sqrt{115}$$
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