Answer:
[tex]\textsf{length}=6x - 1[/tex]
[tex]\textsf{width}=x+3[/tex]
Step-by-step explanation:
Area of a rectangle = length × width
Given area: [tex]A=6x^2+17x-3[/tex]
Therefore, [tex]6x^2+17x-3=\sf length \cdot width[/tex]
To find the length and width, we need to factorize the given expression for area.
To factor a quadratic in the form [tex]ax^2+bx+c[/tex]
- Find 2 two numbers (d and e) that multiply to ac and sum to b
- Rewrite b as the sum of these 2 numbers: d + e = b
- Factorize the first two terms and the last two terms separately, then factor out the comment term.
[tex]6x^2+17x-3 \implies a=6, b=17, c=-3[/tex]
[tex]ac=6 \cdot -3=-18[/tex]
[tex]d+e=17[/tex]
So we are looking for a pair of numbers that multiply to -18 and sum to 17.
Factors of 18: 1, 2, 3, 6, 9, 18
Therefore, the two numbers (d and e) that multiply to -18 and sum to 17 are:
18 and -1
Rewrite [tex]17x[/tex] as [tex]+18x-x[/tex]:
[tex]\implies 6x^2+18x-x-3[/tex]
Factor first two terms and last two terms separately:
[tex]\implies 6x(x+3)-1(x+3)[/tex]
Factor out common term [tex](x+3)[/tex]:
[tex]\implies (6x-1)(x+3)[/tex]
As length > width,
[tex]\textsf{length}=6x - 1[/tex]
[tex]\textsf{width}=x+3[/tex]
