/*! 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 9 Solve the given equation. $$\s... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 9

Solve the given equation. $$\sqrt{x+3}=10$$

Short Answer

Expert verified
x=97

Step by step solution

01

- Isolate the square root

The given equation is \( \sqrt{x+3}=10 \). The square root term \( \sqrt{x+3} \) is already isolated on one side of the equation.
02

- Square both sides of the equation

To eliminate the square root, square both sides of the equation: \( (\sqrt{x+3})^2 = 10^2 \). This simplifies to \( x+3=100 \).
03

- Solve for x

Subtract 3 from both sides to isolate \( x \): \( x + 3 - 3 = 100 - 3 \). This simplifies to \( x=97 \).

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Key Concepts

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

Isolating Variables
Isolating the variable is a crucial step in solving any algebraic equation. It involves arranging the equation so that the variable we are solving for is alone on one side of the equation.
In our example, the equation is given as \( \sqrt{x+3}=10 \). Since the square root term \( \sqrt{x+3} \) is already isolated on the left side of the equation, we don't need to rearrange anything here.
  • Look at both sides of the equation.
  • Ensure that only the variable and its related components are on one side.
  • If there are additional terms, move them to the other side by performing the opposite operation (addition becomes subtraction, multiplication becomes division, etc.).
By isolating the variable, you make it easier to understand and solve the equation.
Square Root
Working with square roots in equations involves understanding how to eliminate them to solve for the variable within them. Here鈥檚 how you can handle it:
In our example \( \sqrt{x+3}=10 \), the square root needs to be removed to isolate \( x \).

  • Square Both Sides: Squaring both sides of the equation nullifies the square root. If you square \( \sqrt{x+3} \), you get \( x+3 \).
  • Perform the Squaring: Applying the squaring, we have \( (\sqrt{x+3})^2 = 10^2 \), which simplifies to \( x+3=100 \).
Now, you have turned a square root equation into a linear one, which is easier to solve.
Remember: Always square both sides equally to maintain the equality.
Algebraic Simplification
Algebraic simplification is the process of making an equation simpler and more manageable. It involves combining like terms and performing basic arithmetic operations.
In our example, after squaring both sides, we have the equation \( x+3=100 \). To solve for \( x \):
  • Subtract to Isolate: We subtract 3 from both sides to isolate \( x \). This gives us \( x+3-3=100-3 \).
  • Simplification: Simplify the equation to \( x=97 \).
In algebra, simplification is key. It makes your equation easier to solve and more straightforward. Retain accuracy by keeping operations balanced on both sides of the equation.

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Most popular questions from this chapter

Solve the following problems algebraically. Be sure to label what the variable represents. Xavier made three investments at \(6.5 \%, 7.6 \%,\) and \(9.2 \% .\) The amount invested at \(7.6 \%\) is \(\$ 1000\) less than the amount invested at \(9.2 \%,\) and the amount invested at \(6.5 \%\) is twice the amount invested at \(7.6 \% .\) If the annual income from the three investments is \(\$ 837,\) how much is invested altogether?

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. $$6(\sqrt{m}-\sqrt{n})-(3 \sqrt{m}+6 \sqrt{n})$$

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. $$2(\sqrt{5}-\sqrt{3})+3(\sqrt{3}-\sqrt{5})$$

Perform the indicated operations. Simplify all answers as completely as possible. Assume that all variables appearing under radical signs are nonnegative. $$\frac{\sqrt{3}}{2+\sqrt{3}}$$

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. $$\sqrt{20}+\sqrt{40}+\sqrt{60}$$
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