Question

The sum of two numbers is 52. Their product is 100. What is the answer?

Answer 1

Answer:

{2, 52}

Step-by-step explanation:

Actually, there are two answers, not just one.

Let the numbers be x and y.  

Then their sum is x + y and equals 52.

Their product is xy and equals 100.

We need to solve this system of equations for both x and y.

Solving x + y = 52 for x, we get x = 52 - y.

Substituting this result into the other equation, we get:

(52 - y)y = 100

This is a quadratic equation:  52y - y² = 100, or

-y² + 52y - 100 = 0.

Alternatively, y² - 52y + 100 = 0

Let's solve this quadratic by "completing the square."

First, identify the coefficient of the y term.  It is -52.  Take half of that, obtaining -26, square this result, obtaining 676.  Add 676 to y² - 52y, and then subtract 676 from the result:

y² - 52y + 676 - 676 + 100 = 0.

Then (y -26)² = 576.

Taking the square root of both sides, we get:

y - 26 = ± 24, or:

y = 24 + 26 = 50, and also:

y = -24 + 26 = 2

Note that 2(50) = 100, as it must, and that

2 + 50 = 52 (also correct).

The two-part answer is {2, 52}.

Answer 2

Answer: 50,2

Explanation:

Let’s call the two numbers a and b.

The sum of them is 52 so a+b=52

The product of them is 100 so ab=100

We now have a system which we can solve in a variety of ways.

My way:

ab=100; b=100/a

a+b=52; a+(100/a)=52; (a^2/a)+(100/a)=52; (a^2+100)/a=52

a^2+100=52a

a^2-52a+100=0

This is now a quadratic which we can solve by factoring. We can use the factors -50 and -2

Therefore, a^2-52a+100=

(a-50)(a-2) or a equals 2 OR 50

Therefore, b=(100/2) or b=(100/50) so b equals 50 OR 2 respectively.

The answers are 50 and 2