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Chapter 0: Problem 136

List all numbers that must be excluded from the domain of each rational expression. $$\frac{7}{2 x^{2}-8 x+5}$$

Short Answer

Expert verified
The numbers that must be excluded from the domain of the rational expression \( \frac{7}{2 x^{2}-8 x+5} \) are \(x_1 = 1 - \sqrt{1.5}\) and \(x_2 = 1 + \sqrt{1.5}\).

Step by step solution

01

Identify the denominator of the rational expression

The denominator of the given rational expression \( \frac{7}{2 x^{2}-8 x+5} \) is \(2 x^{2}-8 x+5 \). This denominator cannot be zero.
02

Set the denominator equal to zero and solve the equation

To identify which values of x make the expression undefined, set the denominator equal to zero: \(2 x^{2}-8 x+5 = 0\). Solving this quadratic equation will give the values of x that must be excluded from the domain.
03

Solve the quadratic equation

To solve the quadratic equation \(2 x^{2}-8 x+5 = 0\), you can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\), where a, b, and c are the coefficients of the quadratic equation. In this case, a = 2, b = -8, and c = 5. Substituting these values into the formula gives: \(x = \frac{8 \pm \sqrt{(-8)^2-4*2*5}}{2*2}\)
04

Calculate the discriminant

The discriminant is the part of the quadratic formula under the square root, which is \(b^2-4ac\). Let's calculate it for this case: \(discriminant = (-8)^2-4*2*5 = 64 - 40 = 24\). Since the discriminant is positive, there will be two different real solutions.
05

Identify the solutions

Substituting the discriminant into the quadratic formula gives two solutions: \(x = \frac{8 \pm \sqrt{24}}{4} = \frac{8 \pm 2\sqrt{6}}{4}\). By simplifying these equations, we have: \(x_1 = 1 - \sqrt{1.5}\) and \(x_2 = 1 + \sqrt{1.5}\). These are the values that we need to exclude from the domain of the original expression.

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Most popular questions from this chapter

Perform the indicated operations. Simplify the result, if possible. $$\left(2-\frac{6}{x+1}\right)\left(1+\frac{3}{x-2}\right)$$

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?

What's wrong with this argument? Suppose \(x\) and \(y\) represent two real numbers, where \(x>y .\) $$\begin{aligned} &2>1\\\ &2(y-x)>1(y-x)\\\ &2 y-2 x>y-x\\\ &\begin{aligned} y-2 x &>-x \\ y &>x \end{aligned} \end{aligned}$$ The final inequality, \(y>x,\) is impossible because we were initially given \(x>y\)

Exercises \(142-144\) will help you prepare for the material covered in the next section. Simplify and express the answer in descending powers of \(x\) : $$ 2 x\left(x^{2}+4 x+5\right)+3\left(x^{2}+4 x+5\right) $$

Explain how to simplify a rational expression.
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