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Chapter 1: Problem 19

Solve each equation. $$\frac{5}{x^{2}}-\frac{43}{x}=18$$

Short Answer

Expert verified
The solutions are \( x = \frac{1}{9} \) and \( x = -\frac{5}{2} \).

Step by step solution

01

Find a Common Denominator

The equation given is \( \frac{5}{x^{2}}-\frac{43}{x}=18 \). Notice that the common denominator of \( x^2 \) and \( x \) is \( x^2 \). Therefore, rewrite the fractions with \( x^2 \) as the denominator: \( \frac{5}{x^{2}} - \frac{43x}{x^2} = 18 \).
02

Combine the Fractions

Combine the fractions to have a single fraction on the left-hand side: \( \frac{5 - 43x}{x^2} = 18 \).
03

Clear the Denominator

Multiply both sides of the equation by \( x^2 \) to eliminate the fraction: \( 5 - 43x = 18x^2 \).
04

Rearrange the Equation

Rearrange the equation to set it to zero: \( 18x^2 + 43x - 5 = 0 \).
05

Solve the Quadratic Equation

Solve the quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \( a = 18 \), \( b = 43 \), and \( c = -5 \): Calculate the discriminant: \( b^2 - 4ac = 43^2 - 4(18)(-5) = 1849 + 360 = 2209 \).Now solve for \( x \) using the quadratic formula: \( x = \frac{-43 \pm \sqrt{2209}}{2(18)} = \frac{-43 \pm 47}{36} \).
06

Simplify the Quadratic Formula

Simplify the expressions \( x = \frac{-43 + 47}{36} = \frac{4}{36} = \frac{1}{9} \) and \( x = \frac{-43 - 47}{36} = \frac{-90}{36} = -\frac{5}{2} \).

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Key Concepts

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

quadratic formula
The quadratic formula is a powerful tool used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). This formula provides the solution(s) for \(x\) as long as you know the coefficients \(a\), \(b\), and \(c\). The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. It鈥檚 important to follow each step carefully:
  • Identify the coefficients \(a\), \(b\), and \(c\) from your equation.
  • Calculate the discriminant \(b^2 - 4ac\).
  • Plug the values into the formula.
  • Simplify to get the solutions for \(x\).
In our example: \(18x^2 + 43x - 5 = 0\), \(a = 18\), \(b = 43\), and \(c = -5\), and from here we can use the quadratic formula to find \(x\).
common denominator
When solving rational equations, finding a common denominator is crucial. This helps to combine fractions and simplify the equation. Here鈥檚 how you can identify and use a common denominator:
  • Look at the denominators involved in each term.
  • Find the lowest common multiple of these denominators.
  • Rewrite each fraction with the common denominator.
In the given problem: \( \frac{5}{x^{2}} - \frac{43}{x} = 18 \), the common denominator is \(x^2\). Rewrite each term to have \(x^2\) as the denominator: \( \frac{5}{x^2} - \frac{43x}{x^2} = 18 \). This simplifies fractions and lets us eliminate the denominators in the next step by multiplying through by \(x^2\).
discriminant
The discriminant is a component of the quadratic formula under the square root sign: \(b^2 - 4ac\). It tells us the nature of the roots (solutions) of the quadratic equation:
  • If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
  • If \(b^2 - 4ac = 0\), the equation has exactly one real root (a repeated root).
  • If \(b^2 - 4ac < 0\), the equation has two complex (imaginary) roots.
In our problem: \(43^2 - 4(18)(-5) = 2209\), since \(2209 > 0\), there are two distinct real-root solutions for \(x\). Calculating further, we find those roots to be \(x = \frac{1}{9}\) and \(x = -\frac{5}{2}\). Understanding the discriminant helps us determine what kind of solutions we should expect before even solving the equation.

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Most popular questions from this chapter

Buying and Selling Land Roger bought two plots of land for a total of \(\$ 120,000\). When he sold the first plot, he made a profit of \(15 \% .\) When he sold the second, he lost \(10 \% .\) His total profit was \(\$ 5500 .\) How much did he pay for each piece of land?

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Solve each rational inequality. Write each solution set in interval notation.4 $$\frac{9 x-8}{4 x^{2}+25}<0$4
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