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Chapter 0: Problem 157

In solving \(\sqrt{2 x-1}+2=x,\) why is it a good idea to isolate the radical term? What if we don't do this and simply square each side? Describe what happens.

Short Answer

Expert verified
In solving the equation, isolating the radical term first is essential. It avoids getting extraneous roots and unnecessary terms in the equation after squaring both sides. The valid root for given equation is \(x = 5\).

Step by step solution

01

Isolating the Radical Term

First, the equation \(\sqrt{2x - 1} + 2 = x\) needs to be rearranged in order to isolate the radical term. This can be done by subtracting 2 from each side, giving the result as \(\sqrt{2x - 1} = x - 2\). By doing this, it's easier to focus on the radical operation.
02

Squaring Both Sides

The next step is to square both sides of the equation. As the radical is now isolated, squaring both sides of the equation will remove the square root, resulting in \(2x - 1 = (x - 2)^2\). This will help in further simplification of the equation.
03

Expanding and Simplifying

Expand the right side of the equation to achieve \(2x - 1 = x^2 - 4x + 4\). Taking all the terms to one side, we get a quadratic equation, \(0 = x^2 - 6x + 5 = (x-5)(x-1)\). Which yields \(x = 5\) and \(x = 1\).
04

Checking the Roots

Checking these solutions in the original equation, \(x = 1\) doesn't satisfy the equation, making this extraneous, meaning it came about due to squaring both sides. Thus the only valid root to this equation is \(x = 5\).
05

Without Isolating the Radical Term

If we square both sides without the radical isolated, the equation becomes \((\sqrt{2x - 1} + 2)^2 = x^2\). With this, extra terms and an additional square root appear, creating a more complex equation to solve, which often may lead to incorrect solutions.

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Most popular questions from this chapter

Will help you prepare for the material covered in the first section of the next chapter. If \(y=4-x^{2},\) find the value of \(y\) that corresponds to values of \(x\) for each integer starting with -3 and ending with 3

$$\text { Solve for } C: \quad V=C-\frac{C-S}{L} N$$.

What is a quadratic equation?

Perform the indicated operations. Simplify the result, if possible. $$\frac{y^{-1}-(y+2)^{-1}}{2}$$

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it costs \(\$ 3,00\) to produce cach package of stationery. The selling price is \(\$ 5.50\) per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?
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