If the expression \frac{4x^2 + kx - 7}{2x - 1} is equivalent to 2x + 5 - \frac{2}{2x - 1} … — SAT

If the expression $\frac{4x^2 + kx - 7}{2x - 1}$ is equivalent to $2x + 5 - \frac{2}{2x - 1}$ for all $x \neq \frac{1}{2}$ , where $k$ is a constant, what is the value of $k$ ?

Answer: 8

Answer

The correct answer is 8. Combining the expression

2x + 5 - \frac{2}{2x - 1}

into a single fraction requires finding a common denominator of

2x - 1

. Multiplying the linear term

2x + 5

\frac{2x - 1}{2x - 1}

and subtracting 2 yields

\frac{(2x + 5)(2x - 1) - 2}{2x - 1}

. Expanding and simplifying the numerator gives

4x^2 + 8x - 7

. Equating this to the numerator of the original expression,

4x^2 + kx - 7

, shows that the coefficient of the linear term,

k

, must be equal to 8.

Step-by-Step Solution

Multiply the linear expression by the denominator to prepare for combining the terms.

(2x + 5)(2x - 1) = 4x^2 + 8x - 5

To combine all terms under a single common denominator of

2x - 1

, the non-fractional terms must be multiplied by the denominator.

Subtract the numerator of the fractional term from the expanded product.

(4x^2 + 8x - 5) - 2 = 4x^2 + 8x - 7

This completes the subtraction of the fraction over the common denominator, resulting in a single rational expression.

Compare the resulting numerator to the numerator of the original expression to find the value of the constant.

k = 8

For the two rational expressions to be equivalent, their numerators must be equal for all values of

x

. Thus, the coefficient of

x

in both expressions must match.

Key Concept

Equivalence of rational expressions through finding a common denominator

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