Find the length of the side of a square if it's area is 60 more than its perimeter.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

Let the length of the side of the square be S.

Now the perimeter is given by 4S and the area if given by S^2.

We cannot say that area is 60 more than the perimeter as they have different units. We can say the same only about their magnitude.

Doing that S^2 = 4S + 60

=> S^2 - 4S - 60 = 0

=> S^2 - 10S + 6S - 60 = 0

=> S(S - 10) + 6( S - 10) = 0

=> (S + 6)(S - 10) = 0

So we have S = -6 or 10.

Now length cannot be negative, so we ignore S = -6.

The length of the side of the square is 10.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll note as x the side of the square.

We'll write the formula for the area of the square:

A = x^2

We'll write the formula for the perimeter of the square:

P = 4x

Now, we'll write mathematically the condition imposed by enunciation:

x^2 - 60 = 4x (area is 60 less than the perimeter)

We'll subtract both sides 4x:

x^2 - 4x - 60 = 4x - 4x

We'll eliminate like terms:

x^2 - 4x - 60 = 0

We'll apply the quadratic formula:

x1 = [4+sqrt(16 + 240)]/2

x1 = (4+16)/2

x1 = 10

x2 = (4-16)/2

x2 = -6

Since the length of the side of the square cannot be negative, we'll reject the second root x2 = -6.

The length of the side of the square is x = 10.

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