Mr. Sanchez's students were asked to add two positive integers. Juan subtracted by mistake and got 2. Maria mistakenly multiplied and got 120. What was the correct answer?

22

Step-by-step explanation:

Given

From Juan's calculation,

Difference of two positive integers = 2

From Maria's calculation,

Product of same integers = 120

Required

Find the sum of the two numbers

Let the two integers be represented by a and b

a - b = 2 (1)

a * b = 120 (2)

Make a the subject of formula in (1)

a = 2 + b

Substitute 2 + b for a in (2)

(2 + b) * b = 120

Open bracket

2 * b + b * b = 120

2b + b² = 120

Rearrange

b² + 2b = 120

Subtract 120 from both sides

b² + 2b - 120 = 120 - 120

b² + 2b - 120 = 0

At this point, we have a quadratic equation.

We start by expanding the expression

b² + 12b - 10b - 120 = 0

Factorize

b(b + 12) - 10(b + 12) = 0

(b - 10)(b + 12) = 0

This implies that

b - 10 = 0 or b + 12 = 0

Make b the subject of formula in both cases

b = 10 or b = -12

From the question, we understand that both numbers are positive.

This means that

b = -12 will be discarded.

Hence, b = 10

Recall that a = 2 + b

Substitute 10 for b

a = 2 + 10

a = 12

This implies that the two numbers are 12 and 10.

Their sum = 12 + 10

Sum = 22

The correct answer is 22