Q10. - Algebra 1

Question

Jorge has ${\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{8}\mathit{x}\mathbf{+}\mathbf{15}$ square yards of carpet. He wants to carpet rooms that have areas of ${\mathit{x}}^{\mathbf{2}}\mathbf{+}\mathbf{5}\mathit{x}\mathbf{+}\mathbf{6}$ square yards. Write and simplify an expression to show how many rooms he can carpet,

Step-by-Step Solution

Answer

Option $\mathit{B}$ is the correct answer.

Step 1. Define an expression.

A mathematical expression is a combinations of numbers and variables ordered with mathematical operations.

Step 2. Define factorization of the expression x 2 + x ( a + b ) + a b .

$x^2+x(a+b)+ab=(x+a)(x+b)$

Step 3. Determine the expression for the number of rooms that Jorge can carpet.

Area of the available carpet$=(x^2+8x+15)$square yards

Area of the room$=(x^2+5x+6)$square yards

To find the number of rooms divide the area of the available carpet by the area of the room.

The number of rooms that Jorge can carpet is 

$=\frac{x^2+8x+15}{x^2+5x+6}$ 

To simplify the expression, Factorise the numerator and the denominator.

$\begin{array}{l}=\frac{x^2+5x+3x+15}{x^2+3x+2x+6}\\ =\frac{x(x+5)+3(x+5)}{x(x+3)+2(x+3)}\\ =\frac{(x+5)(x+3)}{(x+3)(x+2)}\\ =\frac{x+5}{x+2}\end{array}$

Hence, the number of rooms that Jorge can carpet is$\frac{x+5}{x+2}$.

Therefore, option$B$is the correct answer.