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How to Solve the Rational Equation 10x/(x-8) - 32/(x+8) = 512/(x²-64)
Mathematics
Grade 10 (Junior High School)
Question Content
Solve the rational equation: $\frac{10x}{x-8} - \frac{32}{x+8} = \frac{512}{x^2 - 64}$
Correct Answer
x = 8 is an extraneous solution, so the equation has no valid solutions
Detailed Solution Steps
1
Step 1: Identify the common denominator. Notice that $x^2 - 64$ is a difference of squares, which factors to $(x-8)(x+8)$. This is the least common denominator (LCD) of the three fractions.
2
Step 2: Multiply every term in the equation by the LCD $(x-8)(x+8)$ to eliminate the denominators: $10x(x+8) - 32(x-8) = 512$
3
Step 3: Expand and simplify the left-hand side: $10x^2 + 80x - 32x + 256 = 512$, which simplifies to $10x^2 + 48x + 256 = 512$
4
Step 4: Rearrange the equation to standard quadratic form by subtracting 512 from both sides: $10x^2 + 48x - 256 = 0$. Divide all terms by 2 to simplify: $5x^2 + 24x - 128 = 0$
5
Step 5: Solve the quadratic equation using factoring, quadratic formula, or completing the square. Factoring gives $(5x + 40)(x - 8) = 0$, so the potential solutions are $x = -8$ and $x = 8$
6
Step 6: Check for extraneous solutions. Substitute $x=8$ into the original equation: the denominators $x-8$ becomes 0, which is undefined. Substitute $x=-8$: the denominator $x+8$ becomes 0, which is also undefined. Both solutions make the original equation undefined, so there are no valid solutions.
Knowledge Points Involved
1
Rational Equations
Equations that contain one or more rational expressions (fractions with polynomial numerators and denominators). To solve them, eliminate denominators by multiplying by the least common denominator, then solve the resulting polynomial equation. Always check solutions because they may make original denominators zero (extraneous solutions).
2
Difference of Squares
A special factoring pattern: $a^2 - b^2 = (a-b)(a+b)$. Used here to factor $x^2 - 64$ into $(x-8)(x+8)$, which helps find the common denominator for the rational equation.
3
Extraneous Solutions
Solutions obtained when solving rational or radical equations that do not satisfy the original equation. They usually occur because multiplying by an expression containing a variable can introduce solutions that make the original denominator (or radicand) undefined, so all potential solutions must be verified in the original equation.
4
Quadratic Equations
Polynomial equations of degree 2, in the form $ax^2 + bx + c = 0$. They can be solved by factoring, completing the square, or using the quadratic formula $x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$. In this problem, eliminating denominators from the rational equation results in a quadratic equation.
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