Question

write an equation to represent the shaded area. then find the value of x. see example 3 30. total area = 198 cm² image of a rectangle with a smaller rectangle inside, labeled with 9 cm, 16 cm, and x for the shaded regions

Explanation:

Step1: Define total area equation

The total area is the area of the large rectangle (outer shape) which has dimensions $(16+x)$ cm and $(9+x)$ cm, equal to $198$ cm².
$$(16+x)(9+x) = 198$$

Step2: Expand left-hand side

Multiply the binomials using distributive property.
$$144 + 16x + 9x + x^2 = 198$$
$$x^2 + 25x + 144 = 198$$

Step3: Rearrange to standard quadratic form

Subtract 198 from both sides to set equation to 0.
$$x^2 + 25x - 54 = 0$$

Step4: Factor the quadratic

Find two numbers that multiply to -54 and add to 25.
$$(x+27)(x-2) = 0$$

Step5: Solve for x

Set each factor equal to 0, discard negative length.
$$x+27=0 \implies x=-27 \ (\text{invalid})$$
$$x-2=0 \implies x=2$$

Answer:

Equation: $\boldsymbol{(x+16)(x+9)=198}$ (or expanded $\boldsymbol{x^2 +25x -54=0}$)
Value of $x$: $\boldsymbol{2}$ cm