Question

the area of a rectangle is 24 square centimeters. the length (x), in centimeters, of the rectangle is given by the equation ( x^2 + 5x = 24 ). what is the length of the rectangle?
a. 3 cm
b. 5 cm
c. 8 cm
d. 12 cm

Explanation:

Step1: Rearrange the equation

We start with the equation \(x^{2}+5x = 24\). To solve for \(x\), we first rearrange it into a quadratic equation in standard form \(ax^{2}+bx + c=0\). Subtract 24 from both sides:
\(x^{2}+5x - 24=0\)

Step2: Factor the quadratic equation

We need to find two numbers that multiply to \(- 24\) and add up to \(5\). The numbers are \(8\) and \(-3\) because \(8\times(-3)=-24\) and \(8+( - 3)=5\). So we can factor the quadratic as:
\((x + 8)(x-3)=0\)

Step3: Solve for \(x\)

Using the zero - product property, if \((x + 8)(x - 3)=0\), then either \(x+8 = 0\) or \(x - 3=0\).

• If \(x+8=0\), then \(x=-8\). But the length of a rectangle cannot be negative, so we discard this solution.
• If \(x - 3=0\), then \(x = 3\). Wait, but let's check again. Wait, maybe we made a mistake in factoring? Wait, no, let's re - check the equation. Wait, the area is 24, and the equation is \(x^{2}+5x=24\). Wait, maybe the equation is related to length and width. Let's assume that the width is \(x\) and length is \(x + 5\) (since the equation is \(x^{2}+5x\), which is \(x(x + 5)\), so area \(A=x(x + 5)=24\)). So \(x^{2}+5x-24 = 0\). Wait, when we factor \(x^{2}+5x - 24\), we need two numbers that multiply to \(-24\) and add to \(5\). The numbers are \(8\) and \(-3\), so \((x + 8)(x-3)=0\). But if \(x\) is the width, then \(x = 3\) (since width can't be negative), and length is \(x + 5=3 + 5 = 8\)? Wait, no, the problem says "the length \(x\) of the rectangle is given by the equation \(x^{2}+5x = 24\)". Wait, maybe I misinterpreted. Let's solve the quadratic equation \(x^{2}+5x-24=0\) using the quadratic formula \(x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\), where \(a = 1\), \(b = 5\), \(c=-24\). Then \(x=\frac{-5\pm\sqrt{25+96}}{2}=\frac{-5\pm\sqrt{121}}{2}=\frac{-5\pm11}{2}\). So \(x=\frac{-5 + 11}{2}=\frac{6}{2}=3\) or \(x=\frac{-5-11}{2}=\frac{-16}{2}=-8\). But the length can't be negative, so \(x = 3\)? But the options have 3 as option A. Wait, but let's check the area. If \(x = 3\), then from the equation \(x^{2}+5x=9 + 15=24\), which matches the area. So the length is 3 cm? Wait, but maybe the equation is length times width, and if length is \(x\) and width is \(x + 5\), then \(x(x + 5)=24\), so \(x^{2}+5x=24\), and when \(x = 3\), width is \(3\), length is \(8\)? Wait, no, the problem says "the length \(x\) of the rectangle is given by the equation \(x^{2}+5x = 24\)". So we solve for \(x\) and get \(x = 3\) (since \(x=-8\) is invalid). So the length is 3 cm.

Answer:

A. 3 cm