If two integers add up to 35 and their product is 304. What are the numbers?  

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

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The two integers add up to 35 and their product is 304. This is as good as the two integers are the  solution of the quadratic equation given below:

x^2-35x+304 = 0...(1)

As the sum of the roots x1+x2= -(-35 ) = 35

And the product of roots  x1x2 = 304.

We factor the LHS of (1) to get the roots.

x^2 -19x-16x+304 = 0

x(x-19) -16(x-19) = 0

(x-16)(x-19) = 0

x-16 = 0 or x-19 = 0

Therefore x1 = 16 and x2 = 19 are the two numbers whose sum is 35 and product = 304.

william1941's profile pic

william1941 | College Teacher | (Level 3) Valedictorian

Posted on

The two numbers have a sum of 35. Let one of the numbers be x.

Therefore the other number is 35- x

Now we know that the product of the numbers is 304. Therefore (35-x)*x = 304

=> 35x – x^2 = 304

=> x^2  - 35x + 304 =0

=> x^2 – 16x – 19x +304 = 0

=> x(x-16) -19(x -16) +304 =0

=> (x-16)(x-19) = 0

Therefore x can be 16 or 19.

So the required numbers are 16 and 19.

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