Question

the total area of the figure to the right is 209 cm². use this fact to write an equation involving x. then solve the equation to find the value of x. the equation is □.

Explanation:

Step1: Find the dimensions of the large - rectangle

The length of the large rectangle is $(x + 15)$ cm and the width is $(x + 7)$ cm. The area of a rectangle is given by $A = length\times width$. So the area of the large rectangle is $A=(x + 15)(x + 7)$.

Step2: Set up the equation

Since the area of the figure is 209 $cm^2$, we have the equation $(x + 15)(x + 7)=209$.
Expand the left - hand side using the FOIL method: $x^2+7x + 15x+105 = 209$.
Combine like terms: $x^2+22x+105 = 209$.
Subtract 209 from both sides to get a quadratic equation in standard form: $x^2+22x+105 - 209=0$, which simplifies to $x^2+22x - 104 = 0$.

Step3: Solve the quadratic equation

We can solve the quadratic equation $x^2+22x - 104 = 0$ using the quadratic formula $x=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 22$, and $c=-104$.
First, calculate the discriminant $\Delta=b^2 - 4ac=(22)^2-4\times1\times(-104)=484 + 416 = 900$.
Then, $x=\frac{-22\pm\sqrt{900}}{2}=\frac{-22\pm30}{2}$.
We have two solutions:
$x_1=\frac{-22 + 30}{2}=\frac{8}{2}=4$ and $x_2=\frac{-22 - 30}{2}=\frac{-52}{2}=-26$.
Since $x$ represents a length, we discard the negative solution.

Answer:

$x = 4$