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

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The sum of two numbers is 13, and their product is 30. What is the sum  [#permalink]

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New post 08 Mar 2017, 05:16
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The sum of two numbers is 13, and their product is 30. What is the sum of the squares of the two numbers?

(A) –229

(B) –109

(C) 139

(D) 109

(E) 229
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Re: The sum of two numbers is 13, and their product is 30. What is the sum  [#permalink]

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New post 08 Mar 2017, 08:30
vikasp99 wrote:
The sum of two numbers is 13, and their product is 30. What is the sum of the squares of the two numbers?

(A) –229

(B) –109

(C) 139

(D) 109

(E) 229


a + b = 13 ; ab = 30

So, we may say a = 10 and b = 3

Quote:
Sum of the squares of the two numbers


\(a^2 + b^2 = 10^2 + 3^2\)

\(a^2 + b^2 = 109\)

Thus, answer must be (D) 109
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Re: The sum of two numbers is 13, and their product is 30. What is the sum  [#permalink]

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New post 10 Mar 2017, 10:39
vikasp99 wrote:
The sum of two numbers is 13, and their product is 30. What is the sum of the squares of the two numbers?

(A) –229

(B) –109

(C) 139

(D) 109

(E) 229


We can let the numbers be a and b. Thus:

a + b = 13 and ab = 30

Since a = 13 - b, we can substitute 13 - b in the equation ab = 30 and we have:

(13 - b)b = 30

13b - b^2 = 30

b^2 - 13b + 30 = 0

(b - 10)(b - 3) = 0

b = 10 or b = 3

Notice that when b = 10, a = 3, and when b = 3, a = 10. Therefore, the two numbers that have a sum of 13 and a product of 39 are 3 and 10, and the sum of their squares is is 9 + 100 = 109.

Alternate Solution:

Let’s square each side of a + b = 13:

(a + b)^2 = 169

a^2 + 2ab + b^2 = 169

Since ab = 30, we can substitute 2ab = 60 in the last equality:

a^2 + b^2 + 60 = 169

a^2 + b^2 = 109

Answer: D
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Re: The sum of two numbers is 13, and their product is 30. What is the sum  [#permalink]

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New post 26 Jul 2017, 06:17
Solve for (x+y)2=13^2.
We have the above expression from the problem statement without squaring both sides.
Now expanding gives x^2+y^2=169-2xy=169-60[xy=30]=109.
Option D.
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Re: The sum of two numbers is 13, and their product is 30. What is the sum  [#permalink]

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New post 26 Jul 2017, 08:41
vikasp99 wrote:
The sum of two numbers is 13, and their product is 30. What is the sum of the squares of the two numbers?

(A) –229

(B) –109

(C) 139

(D) 109

(E) 229




    • 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.


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Re: The sum of two numbers is 13, and their product is 30. What is the sum   [#permalink] 26 Jul 2017, 08:41
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