Solve the problem.
-The sum of the squares of two numbers is 20. The difference of the two numbers is 2. Find the two numbers.
A)-2 and 4; or -4 and 2
B)2 and 4
C)-4 and -2
D)-4 and -2; or 2 and 4

Question

Quizplus · Accepted Answer

Let x and y be the two numbers. From the problem, we have:
x^2 + y^2 = 20    (1)
x - y = 2          (2)
Squaring equation (2), we get:
x^2 - 2xy + y^2 = 4
Adding this to equation (1), we get:
2x^2 - 2xy + 2y^2 = 24
Simplifying, we get:
x^2 - xy + y^2 = 12  (3)
Now, we can use equation (2) to solve for one of the variables in terms of the other. Solving for x, we get:
x = y + 2
Substituting this into equation (3), we get:
(y + 2)^2 - y(y + 2) + y^2 = 12
Simplifying, we get:
3y^2 + 4y - 4 = 0
Solving for y using the quadratic formula, we get:
y = (-4 ± √52)/6
Simplifying, we get:
y = (-2 ± √13)/3
Substituting these values back into equation (2) to solve for x, we get:
x = (-2 ± √13)/3 + 2 = (4 ± √13)/3
Therefore, the two possible pairs of numbers are: 
(-4/3 + (1/3)√13, -2/3 + (2/3)√13) and 
(4/3 + (1/3)√13, -2/3 - (2/3)√13)
These pairs simplify to: 
(-4, -2 + √13) and (2, 4 - √13)
Therefore, the answer is D: (-4 and -2) or (2 and 4), which are the two possible pairs of numbers that satisfy the given conditions. Let the two numbers be x and y. From the given information, we can form two equations:
x^2 + y^2 = 20
x - y = 2
Rearranging the second equation, we get x = y + 2. Substituting this in the first equation, we get:
(y + 2)^2 + y^2 = 20
Simplifying and solving for y, we get y = -2 or y = -4. Substituting these values in x = y + 2, we get x = 0 or x = 2. Therefore, the two numbers are (-2, 4) or (-4, 2). Hence, the answer is D.

Solve the Problem. -The Sum of the Squares of Two Numbers Is 20