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Chapter 10: Problem 24

Solve the given equation. $$\sqrt{3+x}-8=3$$

Short Answer

Expert verified
x = 118

Step by step solution

01

Isolate the Square Root

Add 8 to both sides of the equation to isolate the square root term.\[\begin{equation}\sqrt{3+x} - 8 + 8 = 3 + 8\end{equation}\]This simplifies to:\[\begin{equation}\sqrt{3+x} = 11\end{equation}\]
02

Eliminate the Square Root

Square both sides of the equation to eliminate the square root.\[\begin{equation}(\sqrt{3+x} )^2 = 11^2\end{equation}\]This simplifies to:\[\begin{equation}3 + x = 121\end{equation}\]
03

Solve for x

Subtract 3 from both sides to solve for x.\[\begin{equation}3 + x - 3 = 121 - 3\end{equation}\]This simplifies to:\[\begin{equation}x = 118\end{equation}\]

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Key Concepts

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

Isolating the Variable
Isolating the variable is the first step in solving many algebraic equations. This means getting the variable by itself on one side of the equation. In the provided exercise, the variable is inside the square root. To isolate it, we start by moving other terms away from the square root.

In the given equation: \[ \sqrt{3+x} - 8 = 3 \] we first add 8 to both sides. This helps us isolate the square root expression. The goal is to simplify the equation step by step until the variable is by itself.

\
Squaring Both Sides
Squaring both sides of an equation is an important step, especially when you encounter square roots. Once we have isolated the square root term, we need to eliminate the square root to solve for the variable inside it. We do this by squaring both sides of the equation.

From the isolated form of the equation: \[ \sqrt{3+x} = 11 \] we square both sides to remove the square root: \[ (\sqrt{3+x})^2 = 11^2 \] This gives us: \[ 3 + x = 121 \] Squaring both sides makes the equation simpler and allows us to proceed to the next step of solving for the variable.
Solving Linear Equations
The final step involves solving the linear equation we obtained after squaring both sides. Linear equations are straightforward and involve equations in the form of ax + b = c.

In our case, we have: \[ 3 + x = 121 \] To solve for x, we need to isolate it. This involves moving constants to the other side of the equation by performing basic arithmetic operations. Here, we subtract 3 from both sides: \[ 3 + x - 3 = 121 - 3 \], simplifying to: \[ x = 118 \]

Thus, we have our solution.

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Most popular questions from this chapter

Verify that \(2+\sqrt{10}\) is a solution to the equation \(x^{2}-4 x-6=0\).

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. $$\sqrt{54 x}-\sqrt{24 x}$$

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

Perform the indicated operations. Simplify all answers as completely as possible. Assume that all variables appearing under radical signs are nonnegative. $$\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}$$
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