Step-by-step explanation:
Area = length × width
Area = 220 m²
length = width + 12 >from statement The length of the garden is 12m more than its width
By using your solution:
(w +12)w = 220
w¹ × w¹ = w² >add the exponents but make sure they have the same base
w × 12 = 21w
This equation came from the original formula of Area of a rectangle which is AREA = LENGTH × WIDTH.
The problem did not indicated the original dimension for length, instead it specifies that length is 12 m more than its width, thus length = width + 12
Now that we have an expression for length, we are going to substitute length = width + 12 to the original formula of Area
w² + 12w - 220 = 0
This is from: w² + 2w = 220. To achieve this, you take note of the sign of 220, since it is positive, we will have it subtracted from the both sides of the equation to form he standard form of quadratic which is ax² + bx + c = 0
w² + 12w -220 = 220 -220
w² + 12w -220 = 0 >this is now a perfect square
Think of a factor of 220 that when subtracted from each other will have a difference of 12.
In this case the factors are 10 and 22
In sign convension:
Since we have + and -, we will asign tge positive one to the biggest number to yield that positive (bx).
(w + 22)(w - 10) = 0
To find the roots
w + 22 = 0 w - 10 = 0
w + 22 - 22 = -22 w - 10 + 10 = 10
w = -22 w = 10
When we are talking of dimensions, neglect the number with negative sign because there is no such thing as negative length or width.
SOLUTION:
(This is done by using conpleting the square method, if you don't want to find the factors )
Area = length × width
220 = (w + 12) × w >distribute w
220 = w² + 12w >do completing the square
220 + (12/2 •1)² = w² + 12w + (12/2 •1)²
220 + (6)² = w² + 12w + (6)²
220 + 36 = w² + 12w + 36
220 + 36 = (w + 6)² >take the square root of both sides
√(w + 6)² = √(220 + 36)
w + 6 = ± 16
w + 6 - 6 = 16 -6 w + 6 - 6 = -16 -6
w = 10m w = -22m
To find the dimension of length neglect the negative value of w:
Area = length × width
220 = l × 10 >divide both sides by 10 to isolate l
220m²/10m = (l × 10)/10
l = 22 m
