Question

Two numbers differ by 6. The sun of their squares is 146. Find the number.

Answer 1

Let the two numbers be x and y.

We know that the two numbers differ by 6, so we can write the equation: x - y = 6. (1)

We also know that the sum of their squares is 146, so we can write the equation: x^2 + y^2 = 146. (2)

To solve this system of equations, we can use substitution.

From equation (1), we can express x in terms of y: x = y + 6.

Substituting this expression for x in equation (2), we have: (y + 6)^2 + y^2 = 146.

Expanding and simplifying: y^2 + 12y + 36 + y^2 = 146.

Combining like terms: 2y^2 + 12y + 36 = 146.

Rearranging the equation: 2y^2 + 12y + 36 - 146 = 0.

Simplifying: 2y^2 + 12y - 110 = 0.

Dividing through the equation by 2: y^2 + 6y - 55 = 0.

Factoring: (y + 11)(y - 5) = 0.

Setting each factor equal to zero:

y + 11 = 0,
y = -11.

y - 5 = 0,
y = 5.

So we have two possible solutions for y: y = -11 or y = 5.

If y = -11, substituting this value in the equation x = y + 6 gives us:

x = -11 + 6,
x = -5.

If y = 5, substituting this value in the equation x = y + 6 gives us:

x = 5 + 6,
x = 11.

Therefore, the two numbers are -5 and -11 or 11 and 5.