The area of a square garden in square feet is 49x^2-14x+1. Find the dimensions of the garden in terms of x.
x^2=49x^2-14x+1
Then solve for x.
48x^2-14x + 1=0
(8x-1)(6x-1)=0
and solve for two values of x that are possible.
To find the dimensions of the square garden in terms of x, we need to equate the given area expression to the formula for the area of a square.
Given:
Area of the square garden = 49x^2 - 14x + 1
The formula for the area of a square is:
Area = side^2
Let's set up the equation:
49x^2 - 14x + 1 = side^2
To find the dimensions, we need to find the value of the side of the square. To do this, we can take the square root of both sides of the equation:
√(49x^2 - 14x + 1) = √(side^2)
Now, we have:
7x - 1 = side
Therefore, the dimensions of the square garden in terms of x are:
Length = 7x - 1 units
Width = 7x - 1 units
To find the possible values of x, let's solve the quadratic equation:
48x^2 - 14x + 1 = 0
Using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a), where a = 48, b = -14, and c = 1
Plugging in these values, we have:
x = (-(-14) ± √((-14)^2 - 4(48)(1))) / (2(48))
x = (14 ± √(196 - 192)) / 96
x = (14 ± √4) / 96
x = (14 ± 2) / 96
Therefore, the two possible values for x are:
x1 = (14 + 2) / 96 = 16 / 96 = 1/6
x2 = (14 - 2) / 96 = 12 / 96 = 1/8
So, the two sets of dimensions for the square garden are:
Length = 7(1/6) - 1 = 7/6 - 1 = 1/6
Width = 7(1/6) - 1 = 7/6 - 1 = 1/6
and
Length = 7(1/8) - 1 = 7/8 - 1 = -1/8 (since the value is negative, it is not meaningful in this context)
Width = 7(1/8) - 1 = 7/8 - 1 = -1/8 (since the value is negative, it is not meaningful in this context)