A homeowner wants to plant a garden in his backyard. He decides to have 375 square feet garden and wants the garden to be 10 feet longer than it is wide. Find the dimensions of the garden.

L=W+10

area= LW=375
375=W(W+10)

w^2+10W-375=0

completing the square

W^2+10w+25=400
(w+5)=20
w=15,L=25

width --- x ft

length -- x+10 ft

solve
x(x+10) = 375
x^2 + 10x - 375 = 0
(x+25)(x-15) = 0
x = 15 or x = a negative

width: 15 ft
length : 25 ft

check: 15x25 = 375

To find the dimensions of the garden, we can set up a system of equations based on the given information.

Let's denote the width of the garden as 'w' feet. According to the problem, the length of the garden is 10 feet longer than the width. So, the length of the garden can be represented as 'w + 10' feet.

The area of a rectangle is calculated by multiplying its length by its width. In this case, the area of the garden is 375 square feet.

So, we can write the equation: w * (w + 10) = 375

To solve this equation, we can start by expanding it: w^2 + 10w = 375

Rearranging the equation into a standard quadratic form: w^2 + 10w - 375 = 0

Now, we have a quadratic equation that we can solve using factoring, completing the square, or the quadratic formula.

Factoring this equation might be a bit challenging in this case. Therefore, to get the dimensions of the garden, let's solve the equation using the quadratic formula:

The quadratic formula is given by: x = (-b ± √(b^2 - 4ac)) / 2a

For our equation: w^2 + 10w - 375 = 0

We have:
a = 1
b = 10
c = -375

Applying the quadratic formula:
w = (-10 ± √(10^2 - 4 * 1 * -375)) / (2 * 1)
w = (-10 ± √(100 + 1500)) / 2
w = (-10 ± √1600) / 2
w = (-10 ± 40) / 2

This gives us two possible solutions for the width of the garden:
1) w = (-10 + 40) / 2 = 30 / 2 = 15 feet
2) w = (-10 - 40) / 2 = -50 / 2 = -25 feet

Since the width of a garden cannot be negative, we discard the second solution.

Therefore, the width of the garden is 15 feet.

To find the length, we can substitute the width back into the expression for the length: w + 10
Length = 15 + 10 = 25 feet

So, the dimensions of the garden are 15 feet by 25 feet.