/*! 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 17 \(5 \sqrt{x}=30\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 17

\(5 \sqrt{x}=30\)

Short Answer

Expert verified
The solution is \(x = 36\).

Step by step solution

01

Isolate the Square Root

To solve the equation \(5 \sqrt{x}=30\), we first want to isolate the square root term. We do this by dividing both sides of the equation by 5. This gives us \(\sqrt{x} = 6\).
02

Solve for x by Removing the Square Root

Once \(\sqrt{x}\) is isolated, we remove the square root by squaring both sides of the equation. This transforms the equation into \(x = 6^2\).
03

Calculate the Square

Now, we calculate the square of 6. That means \(6^2 = 36\). Therefore, \(x = 36\).

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Key Concepts

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

Isolating Terms in Equations
When solving equations, one of the most crucial steps is to isolate the term containing the variable you need to solve for. In our example, the square root term \(5 \sqrt{x} = 30\) needs to be isolated so we can simplify the problem and solve for \(x\). To achieve this, we need to move all other terms away from the variable. Here鈥檚 how:

  • Identify the term that includes the variable. In this case, it is \(5 \sqrt{x}\).
  • To get \( \sqrt{x} \) alone, divide both sides of the equation by 5. This operation lightens the load on the variable side, simplifying it to \(\sqrt{x} = 6\).

This crucial first step allows us to focus solely on the square root term, setting the stage for the next operations in the equation.
Squaring Both Sides
Once the variable is isolated under the square root sign, squaring each side of the equation becomes the way to dissolve the square root and find the variable directly. Using our example, \(\sqrt{x} = 6\), you should square both sides to remove the square root:

  • Take the squared left side: The square of \(\sqrt{x}\) simply gives \(x\), as squares undo square roots.
  • Square the right side: Calculate \(6^2\), finding the expression that equals \(x\).

This method is reliable because it preserves equality, making sure both sides of the equation are transformed similarly.
Calculating Squares
The final step involves performing the simple arithmetic operation of squaring a number. By squaring 6, as derived from \(x = 6^2\), you calculate:

  • Expand \(6^2\): Multiply 6 by itself.
  • Perform the multiplication: \(6 \times 6 = 36\).

These calculations let us conclude that \(x = 36\), solving the equation and accurately identifying the value of \(x\). Knowing how to square numbers efficiently is a crucial skill in various mathematical problems beyond square root equations.

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Most popular questions from this chapter

Evaluate each radical without using a calculator or a table. (Objective 1) $$\sqrt[3]{64}$$

For Problems \(55-70\), rationalize the denominators and simplify. All variables represent positive real numbers. $$ \frac{\sqrt{a}-3}{\sqrt{a}+1} $$

The time \(T\), measured in seconds, that it takes for an object to fall d feet (neglecting air resistance) is given by the formula \(T=\sqrt{\frac{d}{16}}\). Find the times that it takes objects to fall 75 feet, 125 feet, and 5280 feet. Express the answers to the nearest tenth of a second.

Find a rational approximation, to the nearest tenth, for each radical expression. (Objective 2) $$6 \sqrt{2}+14 \sqrt{2}$$

Use a calculator to find a rational approximation of each square root. Express your answers to the nearest hundredth. $$\sqrt{95}$$
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