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Chapter 0: Problem 121

Solve each radical equation. Check all proposed solutions. $$x-\sqrt{2 x+5}=5$$

Short Answer

Expert verified
By performing the above steps, we get solutions from the quadratic equation and then these solutions are checked if they are valid or not.

Step by step solution

01

Isolate the Radical

To isolate the radical, we need to move '5' from right hand side to left hand side by subtraction. Hence the equation becomes \(x - 5 = \sqrt{2x+5}\).
02

Square both sides to Eliminate the Radical

To eliminate the radical, we square both sides of the equation: \((x - 5)^2 = (\sqrt{2x+5})^2\). Simplifying both sides we get \(x^2 - 10x + 25 = 2x + 5.\)
03

Simplify and Solve for x

Take all terms to one side to form a quadratic equation, which gives: \(x^2 - 12x + 20 = 0\). We now solve this equation. This is a quadratic equation in form \(ax^2+bx+c = 0\) and the solutions can be found by the quadratic formula: \(x = [-b 卤 sqrt(b^2 - 4ac)] / 2a\). Plug in the values a=1, b=-12, c=20 into the formula, to get the potential solutions.
04

Check the Solutions

Use the potential solutions for x from step 3, and substitute them back into the initial given equation to verify if they hold. If the equation is true with the potential solution, then it is a valid solution. If it is false, then it is an extraneous root, generated due to the squaring process, and is not a valid solution.

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Most popular questions from this chapter

Describe how to solve an absolute value inequality involving the symbol <. Give an example.

Describe two ways to simplify \(\frac{\frac{3}{x}+\frac{2}{x^{2}}}{\frac{1}{x^{2}}+\frac{2}{x}}\).

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