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Chapter 10: Problem 22

Solve the given equation. $$\sqrt{1+x}-8=4$$

Short Answer

Expert verified
x = 143

Step by step solution

01

Isolate the Square Root

First, add 8 to both sides of the equation to isolate the square root term.\[ \sqrt{1+x} - 8 + 8 = 4 + 8 \]\[ \sqrt{1+x} = 12 \]
02

Square Both Sides

Next, square both sides of the equation to eliminate the square root.\[ (\sqrt{1+x})^2 = 12^2 \]\[ 1+x = 144 \]
03

Solve for x

Finally, subtract 1 from both sides to solve for \( x \).\[ 1 + x - 1 = 144 - 1 \]\[ x = 143 \]

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Key Concepts

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

Isolating the Square Root
To solve an equation involving a square root, we first need to isolate the square root term. The reason for this is simple: to clearly see what we are working with, so we can apply further mathematical operations effectively. Consider the equation \( \sqrt{1+x} - 8 = 4 \). The presence of -8 on the left side complicates things. To isolate the square root, we add 8 to both sides:
\[ \sqrt{1+x} - 8 + 8 = 4 + 8 \]
This simplifies to:
\[ \sqrt{1+x} = 12 \]
Now, we have a clearer equation to work with. The square root term is by itself, making it easier to proceed to the next step.
Squaring Both Sides of an Equation
After isolating the square root, the next step involves eliminating it by squaring both sides of the equation. Squaring is the mathematical operation of raising a number to the power of 2. This operation helps us remove the square root. From our isolated equation:
\[ \sqrt{1+x} = 12 \]
We square both sides:
\[ \left(\sqrt{1+x}\right)^2 = 12^2 \]
The left side simplifies thanks to the inverse operation of squaring and square roots:
\[ 1 + x = 144 \] This step transforms our equation from one involving a square root to a simple linear equation.
Solving for Variables in Algebra
Finally, we are left with a linear equation that is easy to solve for the variable. In our case, the equation is:
\[ 1 + x = 144 \]
To isolate \( x \), we need to get rid of the 1 on the left side by subtracting 1 from both sides of the equation:
\[ 1 + x - 1 = 144 - 1 \]
Simplifying this expression gives us:
\[ x = 143 \]
This tells us that the value of \( x \) is 143. Solving for variables typically involves operations such as addition, subtraction, multiplication, or division. Always remember to perform the same operation on both sides of the equation to maintain equality.

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Most popular questions from this chapter

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