Problem 19 - Precalculus Student Solutions Manual 5th

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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: \( \frac{5}{x^{2}} - \frac{43x}{x^2} = 18 \).

Step 2: Combine the Fractions

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

Step 3: Clear the Denominator

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

Step 4: Rearrange the Equation

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

Step 5: 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: \( x = \frac{-43 \pm \sqrt{2209}}{2(18)} = \frac{-43 \pm 47}{36} \).

Step 6: Simplify the Quadratic Formula

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

Key Concepts

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’s 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’s 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.