/*! 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 20 Solve each equation. $$ 2 \s... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 20

Solve each equation. $$ 2 \sqrt{x}=\sqrt{5 x-16} $$

Short Answer

Expert verified
x = 16

Step by step solution

01

- Square Both Sides

Start by squaring both sides of the equation to eliminate the square roots. \( (2 \, \sqrt{x})^2 = (\sqrt{5x - 16})^2 \)This simplifies to: \[ 4x = 5x - 16 \]
02

- Rearrange the Equation

Rearrange the equation to isolate the variable x on one side. Subtract 5x from both sides: \[ 4x - 5x = -16 \]Simplify to: \[ -x = -16 \]
03

- Solve for x

Multiply both sides of the equation by -1 to solve for x: \[ x = 16 \]

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Key Concepts

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

square roots
Square roots are operations that find a number which, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 * 4 = 16. Square roots are represented by the radical symbol 鈭�. Understanding square roots is crucial because many algebraic equations involve them.
algebraic equations
Algebraic equations are mathematical statements that show the equality between two expressions. They often contain variables, which are symbols representing unknown values. In our example, the equation is \(2 \sqrt{x} = \sqrt{5x - 16}\). Solving algebraic equations involves finding the value of the variables that make the equation true.
isolating variables
Isolating variables means manipulating an equation so that the variable of interest is on one side by itself. This usually involves combining like terms and using arithmetic operations to simplify the equation. In our example, after squaring both sides, we rearrange the equation to get \(-x = -16\), then multiply both sides by -1 to isolate x: \(x = 16\). This step makes it easier to solve for the variable.
squaring both sides
Squaring both sides of an equation is a useful technique when dealing with square roots. By raising both sides of an equation to the power of two, you can eliminate the square roots and simplify the equation. In our example, squaring both sides changes \(2 \sqrt{x} = \sqrt{5x - 16}\) to \(4x = 5x - 16\). It鈥檚 important to follow up by simplifying and solving the now quadratic equation.

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Most popular questions from this chapter

A rectangular yard has a length of \(\sqrt{192} \mathrm{m}\) and a width of \(\sqrt{48} \mathrm{m} .\) Choose the best estimate of its dimensions. Then estimate the perimeter. A. \(14 \mathrm{m}\) by \(7 \mathrm{m}\) B. \(5 \mathrm{m}\) by \(7 \mathrm{m}\) C. \(14 \mathrm{m}\) by \(8 \mathrm{m}\) D. \(15 \mathrm{m}\) by \(8 \mathrm{m}\)

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