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Chapter 1: Problem 22

Solve each equation by factoring. [Hint for Exer cises 19-22: First factor out a fractional power.] $$ 3 x^{7 / 2}-12 x^{5 / 2}=36 x^{3 / 2} $$

Short Answer

Expert verified
The solutions are \(x = 6\) and \(x = -2\).

Step by step solution

01

Factor out the common term

To factor the equation \(3x^{7/2} - 12x^{5/2} = 36x^{3/2}\), start by factoring out the smallest power of \(x\), which is \(x^{3/2}\). This gives us:\[x^{3/2}(3x^2 - 12x) = 36x^{3/2}.\]
02

Simplify the equation

Divide both sides by \(x^{3/2}\) (assuming \(x eq 0\), as division by zero is undefined):\[3x^2 - 12x = 36.\]
03

Rewrite the quadratic equation

Rearrange the equation to form a standard quadratic equation:\[3x^2 - 12x - 36 = 0.\]
04

Factor the quadratic equation

Factor the quadratic expression \(3x^2 - 12x - 36\). First, factor out a 3:\[3(x^2 - 4x - 12) = 0.\]Next, factor the quadratic expression inside the parentheses:\[(x - 6)(x + 2).\]Thus, the equation becomes:\[3(x - 6)(x + 2) = 0.\]
05

Find the solutions

Set each factor equal to zero to solve for \(x\):1. \(x - 6 = 0\) leads to \(x = 6\).2. \(x + 2 = 0\) leads to \(x = -2\).Therefore, the solutions are \(x = 6\) and \(x = -2\).

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Key Concepts

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

Fractional Powers
When dealing with fractional powers, it's important to understand that these represent roots of numbers. For example, the fractional power \( x^{3/2} \) is equivalent to \( (x^{1/2})^3 \) or the square root of \( x \) raised to the power of three. Fractional powers follow the same basic rules of exponents. Here are some key points to remember:
  • A power of \( 1/2 \) is a square root; likewise, \( x^{1/n} \) indicates the n-th root of \( x \).
  • Adding fractional powers works similarly to adding whole number powers: \( x^{a/b} \times x^{c/d} = x^{(ad + bc)/(bd)} \).
  • When you need to factor expressions with fractional powers, focus on identifying the smallest power of \( x \) present in all terms. For example, in the equation from the exercise, \( x^{3/2} \) is the smallest power to factor out.
Recognizing and factoring out common fractional powers can simplify the solving process considerably. This step reduces complexity, making equations easier to handle in subsequent steps.
Quadratic Equations
Quadratic equations are a type of polynomial equation of degree two, usually presented in the standard form as \( ax^2 + bx + c = 0 \). They pop up frequently in many areas of math and science due to their simple yet versatile nature. Here's a breakdown of what you can expect with quadratic equations:
  • The solutions can be found using various methods such as factoring, completing the square, or the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • In the previous steps, we performed factoring, which involves expressing \( ax^2 + bx + c \) as a product of two binomials. This method works well, particularly when the equation is factorable by simple numbers.
  • For instance, from the equation \( 3(x^2 - 4x - 12) = 0 \), we then factored \( x^2 - 4x - 12 \) into \( (x - 6)(x + 2) \), which are its two factors.
Quadratics are fundamental when solving real-world problems, as they capture parabolic paths, from projectiles to profit curves, making them incredibly useful.
Solving Equations
Solving equations involves finding the value(s) of the variable that make the equation true. When equations involve multiple steps, such as factoring or isolating variables, it's crucial to follow logical and methodical procedures:
  • Start by simplifying the equation, removing any complex parts before attempting to isolate the variable.
  • As seen in the exercise, after factoring the quadratic, we set each factor to zero. It's a key step because it takes advantage of the zero product property: if \( ab = 0 \), then \( a = 0 \) or \( b = 0 \).
  • The solutions are achieved by solving these simpler equations which were previously factors, such as \( x - 6 = 0 \) and \( x + 2 = 0 \), leading to \( x = 6 \) and \( x = -2 \).
The core principle behind solving equations is isolating the variable or expression of interest. By simplifying piece by piece, consistent logic helps uncover the solution even in complex equations.

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Most popular questions from this chapter

GENERAL: Newsletters A newsletter has a maximum audience of 100 subscribers. The publisher estimates that she will lose 1 reader for each dollar she charges. Therefore, if she charges \(x\) dollars, her readership will be \((100-x)\). a. Multiply this readership by \(x\) (the price) to find her total revenue. Multiply out the resulting quadratic function. b. What price should she charge to maximize her revenue? [Hint: Find the value of \(x\) that maximizes this quadratic function.]

The intersection of an isocost line \(w L+r K=C\) and an isoquant curve \(K=a L^{b}\) (see pages 18 and 32 ) gives the amounts of labor \(L\) and capital \(K\) for fixed production and cost. Find the intersection point \((L, K)\) of each isocost and isoquant. [Hint: After substituting the second expression into the first, multiply through by \(L\) and factor.] $$ 3 L+8 K=48 \text { and } K=24 \cdot L^{-1} $$

BUSINESS: Isocost Lines An isocost line (iso means "same") shows the different combinations of labor and capital (the value of factory buildings, machinery, and so on) a company may buy for the same total cost. An isocost line has equation $$ w L+r K=\mathrm{C} \quad \text { for } L \geq 0, \quad K \geq 0 $$ where \(L\) is the units of labor costing \(w\) dollars per unit, \(K\) is the units of capital purchased at \(r\) dollars per unit, and \(C\) is the total cost. Since both \(L\) and \(K\) must be nonnegative, an isocost line is a line segment in just the first quadrant. a. Write the equation of the isocost line with \(w=8, \quad r=6, \quad C=15,000\), and graph it in the first quadrant. b. Verify that the following \((L, K)\) pairs all have the same total cost. \((1875,0),(1200,900),(600,1700),(0,2500)\)

Write an equation of the line satisfying the following conditions. If possible, write your answer in the form \(y=m x+b\). Slope \(-2.25\) and \(y\) -intercept 3

Evaluate each expression without using a calculator. $$ 16^{-3 / 4} $$
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