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Chapter 6: Problem 21

Solve each equation. $$\frac{x}{4}-\frac{21}{4 x}=-1$$

Short Answer

Expert verified
The solutions are \(x = -7\) and \(x = 3\).

Step by step solution

01

Find a common denominator

To solve the equation \(\frac{x}{4} - \frac{21}{4x} = -1\), first find a common denominator for the fractions. Here, the common denominator is \(4x\).
02

Rewrite the equation

Rewrite each term with the common denominator \(4x\). The equation becomes \(\frac{x \times x}{4x} - \frac{21}{4x} = -1\), which simplifies to \(\frac{x^2 - 21}{4x} = -1\).
03

Eliminate the denominator

Multiply both sides of the equation by \(4x\) to eliminate the denominator: \(x^2 - 21 = -4x\).
04

Move all terms to one side

Move all terms to one side to form a quadratic equation: \(x^2 + 4x - 21 = 0\).
05

Factor the quadratic equation

Factor the quadratic equation \(x^2 + 4x - 21 = 0\). This equation factors to \((x + 7)(x - 3) = 0\).
06

Solve for x

Set each factor equal to zero and solve for \(x\): \(x + 7 = 0\) gives \(x = -7\), and \(x - 3 = 0\) gives \(x = 3\).
07

Verify the solutions

Substitute \(x = -7\) and \(x = 3\) back into the original equation to verify they are valid solutions. After verification, both values satisfy the equation.

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Key Concepts

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

quadratic equations
A quadratic equation is an equation of the form: \( ax^2 + bx + c = 0 \). In our example, after simplifying and rearranging the original equation, we arrived at the quadratic equation: \( x^2 + 4x - 21 = 0 \). This is a typical quadratic equation where:
  • \( a = 1 \)
  • \( b = 4 \)
  • \( c = -21 \)
To solve a quadratic equation, you can use methods like factoring, completing the square, or applying the quadratic formula. In this particular exercise, factoring is the chosen method, which we'll delve into next.
factoring
Factoring is the process of breaking down an equation into simpler terms (factors) that, when multiplied together, give the original equation. For the quadratic equation \( x^2 + 4x - 21 = 0 \), we look for two numbers that multiply to -21 and add up to 4. Here, those numbers are 7 and -3 because:
  • 7 * (-3) = -21
  • 7 + (-3) = 4
Therefore, \( x^2 + 4x - 21 = (x + 7)(x - 3) \). By setting each factor equal to zero, we solve for \( x \). This yields:
  • \( x + 7 = 0 \) resulting in \( x = -7 \)
  • \( x - 3 = 0 \) resulting in \( x = 3 \)
Factoring makes it simpler to solve quadratic equations and find the solutions.
common denominators
Common denominators make it easier to compare, add, or subtract fractions. In the equation \( \frac{x}{4} - \frac{21}{4x} = -1 \), the fractions have different denominators. To solve this, we first need a common denominator. Here, the common denominator for \( \frac{x}{4} \) and \( \frac{21}{4x} \) is \( 4x \).By rewriting the equation with the common denominator, we get: \( \frac{x \times x}{4x} - \frac{21}{4x} = -1 \), which simplifies to \( \frac{x^2 - 21}{4x} = -1 \). Using a common denominator allows us to combine fractions and eliminate denominators effectively.
verifying solutions
Verifying solutions is a crucial step to ensure the solutions you found actually satisfy the original equation. After solving for \( x \), we got \( x = -7 \) and \( x = 3 \). To verify these results, substitute each solution back into the original equation:
  • For \( x = -7 \): \( \frac{-7}{4} - \frac{21}{4(-7)} = -1 \)
  • For \( x = 3 \): \( \frac{3}{4} - \frac{21}{4(3)} = -1 \)
Both substitutions will return true statements, proving that \( x = -7 \) and \( x = 3 \) are indeed valid solutions. This step is essential to avoid errors and ensure your solutions are correct.

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