Question

Determine the area of the shaded region (grey) in the quadratic form.

Hint: Shaded area = (gray rectangle area) – (white rectangle area)

Answer 1

The area of shaded region is:
`13x^2+6x-15`

Explanation:

We can see in the diagram that

`Length\ of\ gray\ rectangle = l_g = 3x+5\\Width\ of\ gray\ rectangle = w_g = 4x-3`

The area of rectangle is given by:

Area = Length * width

Now for gray rectangle

`A_g = l_g * w_g\\= (3x+5)(4x-3)\\= 3x(4x-3)+5(4x-3)\\= 12x^2-9x+20x-15\\=12x^2+11x-15`

For White Rectangle:

`Length = l = 5-x\\Width = w = x\\A_w = l * w\\A_w = (5-x)(x)\\= 5x-x^2`

Now,

The area of shaded region will be calculated by subtracting the area of white triangle from the gray triangle.

`A_s = A_g - A_w\\= (12x^2+11x-15) - (5x-x^2)\\=12x^2+11x-15-5x+x^2\\=12x^2+x^2+11x-5x-15\\=13x^2+6x-15`

Hence,

The area of shaded region is:
`13x^2+6x-15`