Question

The dimensions of a rectangle can be expressed as X+7 and x. If the area of the rectangle is 78 square inches, find the dimensions of the rectangle. 6 5 Roft

Answer 1

The dimensions of the rectangle are "x+7" and "x", and its area is 78in²

The area of a rectangle is equal to the product of its width and length, following the formula:

`A=wl`

Replace it with the given dimensions of the rectangle:

`78=(x+7)x`

To determine the dimensions of the rectangle, the first step is to determine the value of "x", to do so, you have to simplify the expression obtained above.

-First, distribute the multiplication on the parentheses term:

`\begin{gathered} 78=x\cdot x+7\cdot x \\ 78=x^2+7x \end{gathered}`

-Second, you have to equal the expression to zero. To do so, pass "78" to the right side of the equation by applying the opposite operation to both sides of it:

`\begin{gathered} 78-78=x^2+7x-78 \\ 0=x^2+7x-78 \end{gathered}`

The expression obtained is a quadratic equation, to determine the possible values of x for this expression you have to use the quadratic formula:

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

Where

a is the coefficient of the quadratic term

b is the coefficient of the x-term

c is the constant of the expression

For our equation

a=1

b=7

c=-78

Replace the values and simplify:

`\begin{gathered} x=\frac{-7\pm\sqrt[]{7^2-4\cdot1\cdot(-78)}}{2\cdot1} \\ x=\frac{-7\pm\sqrt[]{49+312}}{2} \\ x=\frac{-7\pm\sqrt[]{361}}{2} \\ x=(-7\pm19)/(2) \end{gathered}`

Next, you have to calculate the addition and subtraction separately:

-Addition:

`\begin{gathered} x=(-7+19)/(2) \\ x=(12)/(2) \\ x=6 \end{gathered}`

-Subtraction:

`\begin{gathered} x=(-7-19)/(2) \\ x=(-26)/(2) \\ x=-13 \end{gathered}`

The possible values of x are 6 and -13, since there cannot be negative side lengths, the value corresponding to the rectangle's side is x=6

The final step, determine the missing side:

`x+7=6+7=13`

The dimensions of the rectangle are 6in and 13in