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Chapter 3: Problem 65

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

Short Answer

Expert verified
The solution is x = 7.

Step by step solution

01

Isolate one of the square roots

Start by isolating one of the square roots on one side of the equation. Move \(\sqrt{x+2}\) to the other side:\[\sqrt{x-3} = 5 - \sqrt{x+2}\]
02

Square both sides

Square both sides of the equation to eliminate the square root on the left side. Remember that you must square both terms inside the parentheses on the right side:\[(\sqrt{x-3})^2 = (5 - \sqrt{x+2})^2\]Simplify this to get:\[x-3 = 25 - 10\sqrt{x+2} + (x+2)\]
03

Simplify the equation

Combine like terms to simplify the right-hand side of the equation:\[x-3 = 27 + x - 10\sqrt{x+2}\]
04

Isolate the remaining square root

Subtract \( x \) from both sides and then isolate the square root term:\[-3 = 27 - 10\sqrt{x+2}\]\[-30 = -10\sqrt{x+2}\]
05

Solve for the square root

Divide both sides by -10 to isolate the square root term:\[\sqrt{x+2} = 3\]
06

Square both sides again

Square both sides to remove the square root:\[(\sqrt{x+2})^2 = 3^2\]\[x+2 = 9\]
07

Solve for x

Subtract 2 from both sides to solve for \( x \):\[x = 7\]
08

Verify the solution

Substitute \( x = 7 \) back into the original equation to verify it satisfies both sides:\[\sqrt{7-3} + \sqrt{7+2} = \sqrt{4} + \sqrt{9} = 2 + 3 = 5\]Since both sides are equal, the solution is correct.

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Key Concepts

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

Isolate Square Roots
When solving radical equations, the first crucial step is to isolate one of the square roots on one side of the equation. This makes it easier to deal with the radicals. For the given problem \(\sqrt{x-3}+\sqrt{x+2}=5\), start by isolating \(\sqrt{x-3}\). Move \(\sqrt{x+2}\) to the other side of the equation:
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Most popular questions from this chapter

Solve and write interval notation for the solution set. Then graph the solution set. $$|x-0.5| \leq 0.2$$

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

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Solve. $$m^{1 / 2}=-7$$
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