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Advanced Math / Nonlinear functions Difficulty: Medium

A rectangle has a length of $x$ units and a width of $left parenthesis x minus 15 right parenthesis$ units. If the rectangle has an area of $76$ square units, what is the value of $x$ ?

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Explanation

Choice B is correct. The area of a rectangle is equal to its length multiplied by its width. Multiplying the given length, $x$ units, by the given width, $left parenthesis x minus 15 right parenthesis$ units, yields $x left parenthesis x minus 15 right parenthesis$ square units. If the rectangle has an area of $76$ square units, it follows that $x (x - 15) = 76$ , or $x^{2} - 15 x = 76$ . Subtracting $76$ from both sides of this equation yields $x^{2} - 15 x - 76 = 0$ . Factoring the left-hand side of this equation yields $(x - 19) (x + 4) = 0$ . Applying the zero product property to this equation yields two solutions: $x equals 19$ and $x = - 4$ . Since $x$ is the rectangle’s length, in units, which must be positive, the value of $x$ is $19$ .

Choice A is incorrect. This is the width, in units, of the rectangle, not the value of $x$ .

Choice C is incorrect and may result from conceptual or calculation errors.

Choice D is incorrect. This is the area, in square units, of the rectangle, not the value of $x$ .