Question

holly has two favorite numbers. if you add her favorite numbers, you get 25. if you multiply her favorite numbers, you get 126. what are her mystery numbers?

Explanation:

Step1: Define variables

Let the two numbers be \( x \) and \( y \). We know that \( x + y = 25 \) and \( x \times y = 126 \). From the first equation, we can express \( y = 25 - x \).

Step2: Substitute into the product equation

Substitute \( y = 25 - x \) into \( x \times y = 126 \), we get \( x(25 - x)=126 \).

Step3: Expand and rearrange the equation

Expanding gives \( 25x - x^{2}=126 \), rearranging to standard quadratic form: \( x^{2}-25x + 126 = 0 \).

Step4: Factor the quadratic equation

We need to find two numbers that multiply to 126 and add up to 25. The factors of 126 are: 1 & 126, 2 & 63, 3 & 42, 6 & 21, 7 & 18, 9 & 14. Among these, 9 and 14 add up to 25. So the quadratic factors as \( (x - 9)(x - 14)=0 \).

Step5: Solve for x

Setting each factor equal to zero: \( x - 9 = 0 \) gives \( x = 9 \); \( x - 14 = 0 \) gives \( x = 14 \).

Step6: Find y

If \( x = 9 \), then \( y = 25 - 9 = 14 \). If \( x = 14 \), then \( y = 25 - 14 = 9 \).

Answer:

The two numbers are 9 and 14.