Exercise 48 Page 31

To find the area of a shaded region, we subtract the white region from the area of the whole. In this case, the whole region is a triangle and the white region is a square. We will need to use those respective area formulas. First let's find the length of the base and height of the large triangle.

Now, to find the length of the base of the triangle, we need to add the two lengths shown. x-3+ x+5=2x+2Likewise, we can add the two lengths we see on the triangle side to get its height. x-3+ x+6=2x+3 Let's put our base and height into the area formula for a triangle to find the area of the whole figure.

A=1/2bh

SubstituteII

b= 2x+2, h= 2x+3

A=1/2( 2x+2)( 2x+3)

▼

Simplify right-hand side

Distr

Distribute 1/2

A = (x+1)(2x+3)

Distr

Distribute x+1

A=2x(x+1)+3(x+1)

Distr

Distribute 2x

A=2x^2+2x+3(x+1)

Distr

Distribute 3

A=2x^2+2x+3x+3

AddTerms

Add terms

A=2x^2+5x+3

From that area, we need to subtract the area of the square with side length x-3. Let's find the area of the square.

A=s^2

Substitute

s= x-3

A=(x-3)^2

ExpandNegPerfectSquare

(a-b)^2=a^2-2ab+b^2

A = x^2 - 2 * 3x + 3^2

CalcPowProd

Calculate power and product

A = x^2 - 6x + 9

Let's subtract that area from the triangle's area. cc &2x^2&+&5x&+&3 -&x^2& -& 6x &+& 9 &x^2 &+&11x&-&6 The area of the shaded region is x^2+11x-6.