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.