To determine the length of the rectangle, we need to solve the equation provided:
[tex]\[
x^2 + 5x = 24
\][/tex]
First, we'll transform this equation into a standard quadratic form:
[tex]\[
x^2 + 5x - 24 = 0
\][/tex]
This is a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], where:
[tex]\[
a = 1, \quad b = 5, \quad c = -24
\][/tex]
To solve this quadratic equation, we use the quadratic formula, which is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Let's substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot (-24)}}{2 \cdot 1}
\][/tex]
First, we calculate the discriminant:
[tex]\[
b^2 - 4ac = 5^2 - 4 \cdot 1 \cdot (-24)
\][/tex]
[tex]\[
b^2 - 4ac = 25 + 96 = 121
\][/tex]
Now, we calculate the two possible solutions for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-5 + \sqrt{121}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-5 + 11}{2} = \frac{6}{2} = 3
\][/tex]
And:
[tex]\[
x = \frac{-5 - \sqrt{121}}{2 \cdot 1}
\][/tex]
[tex]\[
x = \frac{-5 - 11}{2} = \frac{-16}{2} = -8
\][/tex]
We obtain two solutions: [tex]\( x = 3 \)[/tex] and [tex]\( x = -8 \)[/tex]. Since the length of a rectangle cannot be negative, we discard [tex]\( x = -8 \)[/tex].
Thus, the length of the rectangle is [tex]\( x = 3 \)[/tex] centimeters.
The correct answer is:
A. 3 cm
