A Quadratic Word Problem
The length of a rectangular garden is 4 yards more than the width. The area of the garden is 60 square yards. Find the dimensions of the garden.
W = Garden width
H = Garden height
A = Garden area
L = W + 4
A = L * W
60 = ( W + 4 ) * W
60 = W ^ 2 + 4 W Add 4 to both sides
64 = W ^ 2 + 4 W + 4
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Remark :
W ^ 2 + 4 W + 4 = ( W + 2 ) ^ 2
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64 = ( W + 2 ) ^ 2
W + 2 = sqrt ( 64 )
W + 2 = + OR - 8 Subtract 2 to both sides
W + 2 - 2 = + OR - 8 - 2
W = + OR - 8 - 2
Solutions :
W = 8 - 2 = 6
and
W - 8 - 2 = - 10
Width can't be negative so :
W = 6 in
L = W + 4 = 6 + 4 = 10 in
To solve this problem, we can set up a quadratic equation. Let's assume that the width of the rectangular garden is "x" yards.
According to the problem, the length of the garden is 4 yards more than the width. So, the length can be expressed as "x + 4" yards.
The area of a rectangle is given by the formula A = length × width. In this case, the area is 60 square yards. So, we have:
(x + 4) × x = 60
Multiplying out the equation, we get:
x^2 + 4x = 60
Rearranging the equation and setting it equal to zero:
x^2 + 4x - 60 = 0
Now, we have a quadratic equation that we can solve.
To factor this quadratic equation, we need to find two numbers that multiply to give -60 and add up to +4.
After trying out different pairs of factors, we find that the numbers are +10 and -6.
So, the factored form of our equation becomes:
(x + 10)(x - 6) = 0
To solve for "x," we set each factor equal to zero:
x + 10 = 0 or x - 6 = 0
Solving for "x" in each equation, we find:
x = -10 or x = 6
Since the width of a rectangle cannot be negative, we discard -10 as an extraneous solution.
Thus, the width of the garden is 6 yards.
To find the length, we substitute the value of "x" back into the expression for the length:
Length = x + 4 = 6 + 4 = 10 yards
So, the dimensions of the garden are a width of 6 yards and a length of 10 yards.