Respuesta :
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:
[tex]A=wl[/tex]Replace it with the given dimensions of the rectangle:
[tex]78=(x+7)x[/tex]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:
[tex]\begin{gathered} 78=x\cdot x+7\cdot x \\ 78=x^2+7x \end{gathered}[/tex]-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:
[tex]\begin{gathered} 78-78=x^2+7x-78 \\ 0=x^2+7x-78 \end{gathered}[/tex]The expression obtained is a quadratic equation, to determine the possible values of x for this expression you have to use the quadratic formula:
[tex]x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}[/tex]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:
[tex]\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=\frac{-7\pm19}{2} \end{gathered}[/tex]Next, you have to calculate the addition and subtraction separately:
-Addition:
[tex]\begin{gathered} x=\frac{-7+19}{2} \\ x=\frac{12}{2} \\ x=6 \end{gathered}[/tex]-Subtraction:
[tex]\begin{gathered} x=\frac{-7-19}{2} \\ x=\frac{-26}{2} \\ x=-13 \end{gathered}[/tex]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:
[tex]x+7=6+7=13[/tex]The dimensions of the rectangle are 6in and 13in
