Question

The area of a triangle is 60 square inches. The length is x-3, and the width is x + 8. Find the value of x, and the dimensions of a rectangle.

Answer 1

x=7

length= 4

width=15

Explanation

Step 1

we have a rectangle,

Let

$\begin{gathered} \text{length}=x-3 \\ \text{width}=x+8 \\ \text{Area = 60 in}^2 \end{gathered}$

now, the area of a rectangle is given by

$\begin{gathered} \text{Area}=\text{ length(l) }\cdot widht(w) \\ \text{Area}=lw \end{gathered}$

replace

$\begin{gathered} 60in^2=(x-3)(x+8) \\ apply\text{ the distributive property to break the parenthesis} \\ 60=(x-3)(x+8) \\ 60=x^2+8x-3x-24 \\ 60=x^2+5x-24 \\ \text{subtract 60 in both sides} \\ 60-60=x^2+5x-24-60 \\ 0=x^2+5x-84\Rightarrow\text{equation} \end{gathered}$

Step 2

solve the equation :

to solve this equaition, we can use the quadratic formula

remember

$\begin{gathered} \text{if} \\ ax^2+bx+c=0 \\ x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ \end{gathered}$

then, let

a=1

b=5

c=-84

replace to solve for x

$\begin{gathered} x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a} \\ x=\frac{-5\pm\sqrt[]{5^2-4\cdot1\cdot-84}}{2\cdot1} \\ x=\frac{-5\pm\sqrt[]{361}}{2} \\ x=\frac{-5\pm19}{2} \\ \end{gathered}$

so

$\begin{gathered} x=\frac{-5\pm19}{2} \\ x=\frac{-5+19}{2}=\frac{14}{2}=7 \\ \end{gathered}$

the only valid answer is the positive one, so

x=7

Step 3

finally, to find teh length and width, replace the x value

so

$\begin{gathered} \text{length=x-3} \\ \text{length}=7-3 \\ \text{lenght}=4 \end{gathered}$

and

$\begin{gathered} \text{width}=\text{ x+8} \\ \text{width}=7+8 \\ \text{width}=15 \end{gathered}$

I hope this helps you