Problem Solving quadratics

Question

A rectangular paddock has perimeter of 600 m and area 21 600 m^2. Find the dimensions of the paddock.

So far, I've figured out the formula is x(300-x)=21600 and rearranged to 300x-x^2=21600. I'm not really sure if this is correct because in the end I managed to get the answer -145.94 or 145.94 and the correct answer in the back of my textbook is 180 m by 120 m. So I'm a bit confused.

Answer 1

$$P=2(l+b)=600\quad A=lb=21600$$ $$l+21600/l=300\implies l^2-300l+21600=0$$ $$l=\frac{300\pm\sqrt{90000-86400}}{2}=\frac{300\pm\sqrt{3600}}2=\frac{300\pm60}2=180,120$$ $$(l,b)\equiv(180,120)\or(120,180)$$

Answer 2

Perimeter ($P$) is given by: $$P = 2l + 2w = 600\tag{1}$$ for the rectangular paddock, and area ($A$) is given by: $$A = lw = 21600\tag{2}$$ You can solve for l or w by substitution as follows. Rearranging equation $(1)$ gives: $$l=300-w\tag{3}$$ Sub $(3)$ into $(2)$: $$(300-w)w = 21600$$ $$\Rightarrow-w^2+300w-21600 = 0$$ and solve for $w$: $$w=120,180$$ and therefore $l=180, 120$. ie Dimensions are $120m\times180m$ or $180m\times120m$ as required

Answer 3

You are right till $300x-x^2=21600$

Rearranging this equation so that all the terms are on one side and make sign of $x^2$ the first term positive, you shall get $x^2-300x+21600=0$

Apply quadratic formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

You shall get two valid answers for $x$.