Question

a rectangle patio is 9 ft by 6 ft. when the length and width are increased by the same amount, the new...? when solving the equation (9 + x)(6 + x) = 88 using the zero product property, what does the solution ( x ) represent? options: one possible length and one possible width, two possible amounts for the widths, two possible amounts for the lengths, possible amounts by which the dimensions were changed

Explanation:

Step 1: Expand the equation

We start with the equation \((9 + x)(6 + x)=88\). First, expand the left - hand side using the FOIL method. \((a + b)(c + d)=ac+ad+bc+bd\), so \((9 + x)(6 + x)=9\times6+9x + 6x+x^{2}=54 + 15x+x^{2}\). The equation becomes \(x^{2}+15x + 54=88\).

Step 2: Rearrange into standard quadratic form

Subtract 88 from both sides of the equation to get it in the form \(ax^{2}+bx + c = 0\). So \(x^{2}+15x+54 - 88=0\), which simplifies to \(x^{2}+15x - 34 = 0\).

Step 3: Solve the quadratic equation

We can solve the quadratic equation \(x^{2}+15x - 34 = 0\) by factoring (or using the quadratic formula). Let's try factoring. We need two numbers that multiply to \(- 34\) and add up to \(15\). The numbers are \(17\) and \(- 2\) since \(17\times(-2)=-34\) and \(17+( - 2)=15\). So the equation factors as \((x + 17)(x-2)=0\).

Step 4: Find the solutions

Using the zero - product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\). So \(x+17 = 0\) gives \(x=-17\), and \(x - 2=0\) gives \(x = 2\). Since \(x\) represents the amount by which the length and width are increased, a negative value for \(x\) (i.e., \(x=-17\)) does not make sense in the context of increasing the dimensions. So we take \(x = 2\). The question is about what the solution \(x\) represents. The original length is \(9\) ft and width is \(6\) ft, and we are increasing both by \(x\). So \(x\) represents the possible amounts by which the dimensions were changed.

Answer:

The solution \(x\) represents the possible amounts by which the dimensions were changed.