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Chapter 1: Problem 45

Solve each equation. $$\sqrt{x+7}+3=\sqrt{x-4}$$

Short Answer

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Step by step solution

01

Isolate the square root terms

First, move the constant term on the left side of the equation to the right side. Subtract 3 from both sides of the equation: \[ \sqrt{x+7} = \sqrt{x-4} - 3 \]
02

Square both sides of the equation

Square both sides to eliminate the square roots: \[ (\sqrt{x+7})^2 = (\sqrt{x-4} - 3)^2 \]This simplifies to: \[ x+7 = (x-4) + 9 - 6\sqrt{x-4} \]Which further simplifies to: \[ x+7 = x-4 + 9 - 6\sqrt{x-4} \]Finally, simplify this to: \[ x+7 = x+5 - 6\sqrt{x-4} \]
03

Simplify the equation

Subtract \(x + 5\) from both sides of the equation: \[ x + 7 - (x+5) = -6\sqrt{x-4} \]This simplifies to: \[ 2 = -6\sqrt{x-4} \]
04

Isolate the square root term

Divide both sides by -6 to isolate the square root term: \[ \frac{2}{-6} = \sqrt{x-4} \]This simplifies to: \[ -\frac{1}{3} = \sqrt{x-4} \]
05

Verify the solution

Since the square root of any non-negative number is always non-negative, having a negative value \(-\frac{1}{3}\) is not valid. Thus, there is no solution for this equation.

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Key Concepts

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

isolating square roots
When solving equations with square roots, the first step is to isolate the square root term. This makes the equation easier to work with.
In the given problem, we start with the equation \( \sqrt{x+7} + 3 = \sqrt{x-4} \).
We want to isolate \( \sqrt{x+7} \) on one side. To do this, we subtract 3 from both sides:
\[ \sqrt{x+7} = \sqrt{x-4} - 3 \]
By isolating the square root term, we prepare the equation for the next step, which is to eliminate the radical.
eliminating radicals
After isolating the square root term, the next step is to eliminate the radical by squaring both sides of the equation. This helps remove the square roots, transforming the equation into a form that is easier to solve.
Consider our equation from the previous step:
\[ \sqrt{x+7} = \sqrt{x-4} - 3 \]
We square both sides:
\[ \left( \sqrt{x+7} \right)^2 = \left( \sqrt{x-4} - 3 \right)^2 \]
Simplifying both sides, we get:
\[ x+7 = (x-4) + 9 - 6\sqrt{x-4} \]
Which further simplifies to:
\[ x+7 = x+5 - 6\sqrt{x-4} \]
By squaring both sides, the radicals disappear, and we are left with a simpler equation to solve.
checking solutions
It is crucial to check potential solutions in radical equations because squaring both sides can introduce extraneous solutions. An extraneous solution is a solution that does not satisfy the original equation.
After isolating and eliminating the radicals, we are left with:
\[ -6\sqrt{x-4} = 2 \]
Dividing both sides by -6, we obtain:
\[-\frac{1}{3} = \sqrt{x-4} \]
However, \sqrt{x-4}\ cannot be negative, since the square root of any number is always non-negative. This means \-\frac{1}{3}\ is not a valid solution. Thus, the equation has no solution.
Always verify your solutions to ensure they satisfy the original equation. This helps catch and discard any extraneous solutions.

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Most popular questions from this chapter

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