The sum of two numbers is 13, and their product is 30. What is the sum

• We are given that \(x + y = 13\) and \(xy = 30\)
• We need to find the sum of squares of the two numbers.

o That is – \(x^2 + y^2\)
• There are two ways to solve this question. Let me discuss the first one.

Method 1

• If we know the formula of \((x+y)^2\), we can easily solve this question under a minute.

o Since \((x+y)^2 = x^2 + y^2 + 2xy\)

o We know the value of \((x+y)\) and \(xy\), if we substitute it in the above equation we will get the value of \(x^2 + y^2\)


 \((13)^2 = x^2 + y^2 + 2*30\)

 \(x^2 + y^2 = 169 – 60 = 109\)

• Thus, the correct answer is Option D.

Method 2

• The sum of two numbers is 13 and their product is 30
• Since the sum is positive and even the product is positive we can conclude that both x and y should be positive.

o Keeping this in mind, let us focus on \(x*y = 30\)

 \(x*y = 6*5 = 3*10 = 15*2\)
 Out of the three cases above only \(x *y = 3* 10\) gives us \(x + y = 3 + 10 = 13\)
o Thus, the value of \(x^2 + y^2 = 10^2 + 3^2 = 100 + 9 = 109\)
• And the correct answer is Option D.