Problem 18 - College Algebra

Question

Solve the quadratic equation by factoring. $$ \frac{x}{3}-\frac{9}{x}=2 $$

Step-by-Step Solution

Answer

The solutions are \( x = 9 \) and \( x = -3 \).

Step 1: Eliminate the Fractions

To eliminate fractions, multiply the entire equation by 3x to get rid of the denominators. This yields:\[ x^2 - 27 = 6x \]

Step 2: Rearrange Into Standard Quadratic Form

Move all terms to one side of the equation to have it in the standard quadratic form:\[ x^2 - 6x - 27 = 0 \]

Step 3: Factor the Quadratic

Look for two numbers that multiply to -27 and add to -6. These numbers are -9 and 3, so we factor the quadratic as:\[ (x - 9)(x + 3) = 0 \]

Step 4: Solve the Factors

Set each factor equal to zero and solve for x:\[ x - 9 = 0 \] gives \( x = 9 \)\[ x + 3 = 0 \] gives \( x = -3 \)

Step 5: Verify the Solutions

Substitute \( x = 9 \) and \( x = -3 \) back into the original equation to check if they are valid solutions. Both values satisfy the original equation, confirming they are correct solutions.

Key Concepts

Factoring

Factoring is a foundational technique used to simplify expressions and solve equations. When it comes to quadratic equations, factoring involves expressing the equation in the form of two binomials that multiply to give the original quadratic. For example, if we have a quadratic equation like \(x^2 - 6x - 27 = 0\), we look for two numbers that multiply to \(-27\) and add to \(-6\). These numbers are \(-9\) and \(3\), allowing us to write the equation as \((x - 9)(x + 3) = 0\). This shows that the equation equals zero when either factor is zero, enabling us to solve for the variable \(x\).

Standard Quadratic Form

The standard quadratic form of an equation is written as \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. This form is critical because it provides a clear structure to apply various methods of solving, such as factoring, completing the square, or using the quadratic formula. In our problem, the equation \(x^2 - 27 = 6x\) was rearranged into the standard quadratic form \(x^2 - 6x - 27 = 0\). By moving all terms to one side of the equation, it allows for straightforward application of solving methods. This rearrangement is an essential step, ensuring the equation is ready for further analysis or solving steps.

Solving Equations

To solve equations, especially quadratics, factoring is a very effective method. Once a quadratic is factored, setting each factor equal to zero lets us solve for the variable's possible values. For example, with \((x - 9)(x + 3) = 0\), set each factor equal to zero:

• \(x - 9 = 0\), which gives \(x = 9\)
• \(x + 3 = 0\), which gives \(x = -3\)

These solutions are found by simply isolating \(x\). The method relies on the property that if a product of numbers equals zero, then at least one of the numbers must be zero. By identifying and solving these factors, we can determine the values of \(x\) that satisfy the quadratic equation.

Verification of Solutions

Verification of solutions is an important step in problem-solving. It ensures that the solutions obtained are correct and satisfy the original equation. In our example, the solutions \(x = 9\) and \(x = -3\) were found. To verify, substitute these back into the original equation \(\frac{x}{3} - \frac{9}{x} = 2\):

• For \(x = 9\), substitute and simplify to see if both sides equal 2.
• For \(x = -3\), perform the same substitution and simplification.

If both values satisfy the equation, this confirms their validity. This step prevents potential errors from unnoticed mistakes during calculations and confirms that our solution process was correct.