The length of a rectangular garden is two less than thrice its width. How long is the garden if the area is 65m²?
Please help
w(3w-2) = 65
what are the factors of 65?
There are no given factors
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To find the length of the garden, let's first assign variables to the given information:
Let's say the width of the garden is "w" meters.
According to the given information: length = 3w - 2
The area of a rectangle is equal to the length multiplied by the width. In this case, the area is given as 65m². So we can set up the following equation:
Area = Length × Width
65 = (3w - 2) × w
Now, we can solve this equation to find the value of "w" (the width of the garden). After finding the width, we can substitute it back into the equation length = 3w - 2 to find the length of the garden.
Let's solve the equation step by step:
Step 1: Expand the equation.
65 = 3w² - 2w
Step 2: Rearrange the equation in standard quadratic form.
3w² - 2w - 65 = 0
Step 3: Solve the quadratic equation to find the possible values of "w".
We can either factor the equation or use the quadratic formula. Let's use the quadratic formula:
The quadratic formula is given by:
w = (-b ± sqrt(b² - 4ac)) / (2a)
For our equation, a = 3, b = -2, and c = -65.
Substituting the values into the quadratic formula, we get:
w = (-(-2) ± sqrt((-2)² - 4(3)(-65))) / (2(3))
w = (2 ± sqrt(4 + 780)) / 6
w = (2 ± sqrt(784)) / 6
w = (2 ± 28) / 6
Now, let's calculate the two possible values of "w":
w₁ = (2 + 28) / 6 = 30 / 6 = 5
w₂ = (2 - 28) / 6 = -26 / 6 = -13/3
Since the width cannot be negative, we discard the negative value. Therefore, the width of the garden is 5 meters.
Now, let's find the length of the garden using the equation length = 3w - 2:
Length = 3(5) - 2
Length = 15 - 2
Length = 13
Therefore, the length of the garden is 13 meters.