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Chapter 8: Problem 37

Solve each radical equation. $$x=\sqrt{2 x-2}+1$$

Short Answer

Expert verified
The solution for this equation is \(x=3\).

Step by step solution

01

Isolate the radical term

Firstly, it is important to isolate the term with the square root on one side of the equation. Original equation \(x=\sqrt{2x-2}+1\) can be written as \(x-1=\sqrt{2x-2}\) by subtracting 1 from each side.
02

Square both sides

Next step is to get rid of the square root, in order to do this, square both sides. This results in \((x-1)^2 = (2x-2)\)
03

Expand and simplify

Expand \((x-1)^2\) and simplify the resulting equation gives \(x^2 - 2x + 1 = 2x - 2\). Now, let's rearrange it to get a quadratic equation. Result is \(x^2 - 4x + 3 = 0\)
04

Solve the quadratic equation

Solve the quadratic equation using the standard methods (i.e., factoring, quadratic formula, etc). Here, the equation \(x^2 - 4x + 3 = 0\) can be factored as \((x-3)(x-1) = 0\)
05

Find the roots/solutions

Now, set each factor equal to zero to solve, giving \(x-3=0 -> x=3\) and \(x-1=0 -> x=1\)
06

Check the solutions

It is imperative to check each solution in the original equation because squaring both sides (step 2) can introduce extraneous solutions. Substitute \(x=1\) and \(x=3\) into the original equation separately. After doing this, it can be concluded that \(x=3\) is a valid solution whereas \(x=1\) is not, since it does not satisfy the original equation.

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Most popular questions from this chapter

$$\text { Simplify: } \quad \sqrt{2}+\sqrt{\frac{1}{2}}$$

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. There's no question that \((-64)^{\frac{1}{3}}=-64^{\frac{1}{3}},\) so I can conclude that \((-64)^{\frac{1}{2}}=-64^{\frac{1}{2}}\).

Simplify each expression. Write answers in exponential form with positive exponents only. Assume that all variables represent positive real numbers. $$\left(\frac{x^{\frac{2}{5}}}{x^{\frac{6}{5}} \cdot x^{\frac{3}{5}}}\right)^{5}$$

Square the real number \(\frac{2}{\sqrt{3}} .\) Observe that the radical is eliminated from the denominator. Explain whether this process is equivalent to rationalizing the denominator.

In Exercises \(53-74\), rationalize each denominator. Simplify, if possible. $$\frac{15}{\sqrt{7}+2}$$
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