Question

solve by factoring.
1
a billboard has an area of $x^2 - 22x + 72$ square meters. what are the dimensions of the billboard?
2
a gym floor has an area of $x^2 + 5x - 36$ square feet. is it possible for one of the dimensions to be $(x - 9)$ feet? explain.

Explanation:

Sub - Question 1

Step 1: Factor the quadratic expression

We have the quadratic expression \(x^{2}-22x + 72\). We need to find two numbers that multiply to \(72\) and add up to \(- 22\). The numbers are \(-18\) and \(-4\) since \((-18)\times(-4)=72\) and \(-18+( - 4)=-22\).
So, \(x^{2}-22x + 72=(x - 18)(x - 4)\)

Step 2: Determine the dimensions

Since the area of a rectangle (billboard is assumed to be rectangular) is length times width, and the area is given by the factored form \((x - 18)(x - 4)\), the dimensions of the billboard are \((x - 18)\) meters and \((x - 4)\) meters.

Step 1: Try to factor the quadratic expression \(x^{2}+5x - 36\) with \((x - 9)\) as a factor

If \((x - 9)\) is a factor, then when we divide \(x^{2}+5x - 36\) by \((x - 9)\) or try to factor it as \((x - 9)(x + a)\), we can find \(a\) by multiplying \((x - 9)(x + a)=x^{2}+(a - 9)x-9a\).
We want \(x^{2}+(a - 9)x-9a=x^{2}+5x - 36\). So, we set up the equations:
\(a - 9 = 5\) (coefficient of \(x\)) and \(-9a=-36\) (constant term).
From \(-9a=-36\), we get \(a = 4\). But if \(a = 4\), then \(a - 9=4 - 9=-5
eq5\).
Alternatively, we can factor \(x^{2}+5x - 36\) directly. We need two numbers that multiply to \(-36\) and add up to \(5\). The numbers are \(9\) and \(-4\) since \(9\times(-4)=-36\) and \(9+( - 4)=5\). So, \(x^{2}+5x - 36=(x + 9)(x - 4)\)

Step 2: Compare with \((x - 9)\)

The factored form is \((x + 9)(x - 4)\), not \((x - 9)\) times another binomial. So, it is not possible for one of the dimensions to be \((x - 9)\) feet.

Answer:

The dimensions of the billboard are \((x - 18)\) meters and \((x - 4)\) meters.

Sub - Question 2