Question

this question has two parts. first, answer part a. then, answer part b.
part a
garden a rectangular garden is 12 feet across and 16 feet long. it is surrounded by a border of mulch that is a uniform width x. the maximum area for the garden, plus border, is 285 square feet.
image omitted
a. write a polynomial equation to represent the situation.
blankx² + blankx + blank = 285
part b
b. write and solve a system of equations.
y = blankx² + blankx + blank, y = blank; x = blank ft
c. what are the dimensions of the garden plus border? list the dimensions from least to greatest.
blank ft by blank ft

Explanation:

Step1: Find total length/width with border

Total width: $12 + 2x$; Total length: $16 + 2x$

Step2: Expand area equation

$$(12+2x)(16+2x) = 285$$
$$192 + 24x + 32x + 4x^2 = 285$$
$$4x^2 + 56x + 192 - 285 = 0$$
$$4x^2 + 56x - 93 = 285 - 285$$
Simplify to: $4x^2 + 56x - 93 = 0$ (rearranged to match given form: $4x^2 + 56x + (-93) = 285$)

Step3: Set up system for Part B

System: $y = 4x^2 + 56x - 93$, $y = 0$
Solve via quadratic formula $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ where $a=4, b=56, c=-93$:
$$x=\frac{-56\pm\sqrt{56^2-4(4)(-93)}}{2(4)}$$
$$x=\frac{-56\pm\sqrt{3136+1488}}{8}$$
$$x=\frac{-56\pm\sqrt{4624}}{8}=\frac{-56\pm68}{8}$$
Take positive root: $x=\frac{12}{8}=1.5$

Step4: Calculate total dimensions

Total width: $12+2(1.5)=15$ ft; Total length: $16+2(1.5)=19$ ft

Answer:

Part A

$4x^2 + 56x + (-93) = 285$

Part B

b. $y = 4x^2 + 56x - 93$, $y = 0$; $x = 1.5$ ft
c. 15 ft by 19 ft