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Chapter 0: Problem 134

Solve each equation. $$\sqrt{x+5}-\sqrt{x-3}=2$$

Short Answer

Expert verified
The solution to the equation is \(x = 3.0625\).

Step by step solution

01

Isolate One of the Square Roots

First, isolate one of the square roots on one side of the equation. In this case, let's move \( \sqrt{x-3} \) to the right side of the equation: \( \sqrt{x+5} = \sqrt{x-3} + 2 \).
02

Square Both Sides of the Equation

To eliminate the square roots, square both sides of the equation. Squaring the left side we get \(x+5\), and on the right side we get \((\sqrt{x-3} + 2)^2\) which simplifies to \(x-3 + 4\sqrt{x-3} + 4\). This gives us the equation \(x + 5 = x - 3 + 4\sqrt{x-3} + 4\).
03

Simplify the Equation

By simplifying, we can isolate the square root term on one side of the equation. Subtract 'x' and '4' from both sides to get: \(1 = 4\sqrt{x-3}\)
04

Divide by 4 on both Sides

Divide entire equation by 4 so we have \( 0.25 = \sqrt{x-3} \).
05

Remove the Square Root

Again, square both sides of the equation to eliminate the square root. This leaves us with \( 0.0625 = x - 3 \)
06

Solve for 'x'

Finally, solve the equation for 'x' by adding '3' to both sides to move the constant on the right to the left. This gives us \(x = 3.0625\)

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