The length of a rectangle is 6x feet and its width is 3x feet. The area of the rectangle is 420 - 6x square feet.

What is the perimeter of the rectangle, in feet?

Hint:
Create a true equation using the fact that LW = Area.
Solve the equation for x by factoring.
Find the length and width.
Calculate the perimeter.
So first i did
(6x)(3x)=420-6x
18x^2=420-6x~ Then what do i do after or what did i do wrong?

18x^2=420-6x

You now have a quadratic equation

18x^2 + 6x - 420 = 0
3x^2 + x - 70 = 0
(x + 5)(3x - 14) = 0
x = -5 or x = 14/3
since x> 0, x = 14/3

so the sides are 6(14/3) or 28
and 3(14/3) or 14

so the perimeter is 2(28+14) = 84 ft

check:
18x^2 = 18(14/3)^2 = 392
420 - 6x = 420 - 6(14/3) = 420 - 28 = 392

You're on the right track! To continue solving for x, let's rearrange the equation to bring all the terms to one side.

18x^2 + 6x - 420 = 0

Next, we can factor out a common factor from the equation to simplify it. In this case, 6 is a common factor.

6(3x^2 + x - 70) = 0

Now, we have a quadratic equation in the parentheses. We can solve for x by factoring or using the quadratic formula. To factor the quadratic equation, we need to find two numbers whose product is -70 and whose sum is 1 (the coefficient of x).

The numbers 10 and -7 fit these requirements, so we can factor the quadratic equation as follows:

6(3x - 7)(x + 10) = 0

Now, we have two possible solutions for x:

1) 3x - 7 = 0 or 2) x + 10 = 0

1) Solving for x in the first equation:

3x - 7 = 0
3x = 7
x = 7/3

2) Solving for x in the second equation:

x + 10 = 0
x = -10

We have two possible values for x: 7/3 and -10. However, since the length and width of a rectangle cannot be negative, we discard the negative value of x.

Therefore, x = 7/3.

To find the length and width of the rectangle, substitute x back into the corresponding expressions:

Length = 6x = 6 * (7/3) = 42/3 = 14 feet
Width = 3x = 3 * (7/3) = 21/3 = 7 feet

Now that we have the measurements of the rectangle (length = 14 feet and width = 7 feet), we can calculate the perimeter.

Perimeter = 2 * (Length + Width) = 2 * (14 + 7) = 2 * 21 = 42 feet

Therefore, the perimeter of the rectangle is 42 feet.