Question

the area, a, of a rectangle is (120x^2 + 78x - 90), and the length, (l), of the rectangle is (12x + 15). which of the following gives the width, (w), of the rectangle? (9x + 4) (10x - 19) (10x - 6) (8x - 6) (with radio buttons and done button)

Explanation:

Step1: Recall area formula for rectangle

The area \( A \) of a rectangle is given by \( A = l \times w \), where \( l \) is the length and \( w \) is the width. So, to find the width \( w \), we can use \( w=\frac{A}{l} \).

Step2: Substitute the given expressions

We know that \( A = 120x^{2}+78x - 90 \) and \( l = 12x + 15 \). So we need to perform polynomial division or factor the numerator to simplify \( \frac{120x^{2}+78x - 90}{12x + 15} \).

First, let's factor the numerator and the denominator:

Factor the numerator:
We can factor out a common factor of 6 from \( 120x^{2}+78x - 90 \):
\( 120x^{2}+78x - 90=6(20x^{2}+13x - 15) \)

Now, factor the quadratic \( 20x^{2}+13x - 15 \). We need two numbers that multiply to \( 20\times(- 15)=-300 \) and add up to 13. The numbers are 25 and - 12.
So, \( 20x^{2}+13x - 15=20x^{2}+25x-12x - 15 = 5x(4x + 5)-3(4x + 5)=(5x - 3)(4x + 5) \)

So the numerator becomes \( 6(5x - 3)(4x + 5) \)

Factor the denominator \( 12x + 15 \): we can factor out 3, so \( 12x + 15 = 3(4x + 5) \)

Now, substitute back into the fraction:
\( \frac{6(5x - 3)(4x + 5)}{3(4x + 5)} \)

Step3: Simplify the fraction

Cancel out the common factors \( 3 \) and \( (4x + 5) \) (assuming \( 4x+5
eq0 \)):
\( \frac{6(5x - 3)(4x + 5)}{3(4x + 5)}=\frac{6}{3}\times(5x - 3)\times\frac{(4x + 5)}{(4x + 5)} = 2(5x - 3)=10x-6 \)

Answer:

\( 10x - 6 \) (corresponding to the option "10x - 6")