In the following questions you must find an answer without referring to choices:

5. If x + y = 7 and xy = 4, then find the value of x² + y².

Correct answer:

41: Right!
But don't try to solve for both x and y. You are not asked to do that—you are asked to find the value of x² + y² so try to manipulate the equations to get that quantity.
Use Strategy 4, p. 80; BASIC SKILL - Mini Math Refresher - Algebra, p. 156; Math Refresher (409), p. 246: Remember and use classic forms, like
(x + y)² = x² + 2xy + y².

We have (1) (x + y)² = x² + 2xy + y² = 7 x 7 = 49, and since xy = 4, we get 2xy = 8.
Substituting 2xy = 8 in Equation (1), we get (x + y)² = x² + 2xy + y² = x² + 8 + y² = 49. Subtracting 8 from both sides of the last equation, we get,
x² + y² = 41.

Incorrect.

Did you answer 49? Did you multiply x + y = 7 by itself and get 49, and think that you got x² + y² = 49?
Use Strategy 4, p. 80; BASIC SKILL - Mini Math Refresher - Algebra, p. 156; Math Refresher (409), p. 246: Remember and use classic forms, like
(x + y)² = x² + 2xy + y².

Don't try to solve for both x and y. You are not asked to do that—you are asked to find the value of x² + y², so try to manipulate the equations to get that quantity.

We have (1) (x + y)² = x² + 2xy + y² = 7 x 7 = 49 and since xy = 4, we get 2xy = 8.
Substituting 2xy = 8 in Equation (1), we get
(x + y)² = x² + 2xy + y² = x² + 8 + y² = 49. Subtracting 8 from both sides of the last equation, we get
x² + y² = 41.

No. Use Strategy 4, p. 80; BASIC SKILL - Mini Math Refresher - Algebra, p. 156; Math Refresher (409), p. 246: Don't try to solve for both x and y. You are not asked to do that—you are asked to find the value of x2 + y2, so try to manipulate the equations to get that quantity.

We have (1) (x + y)² = x² + 2xy + y² = 7 x 7 = 49 and since xy = 4, we get 2xy = 8.
Substituting 2xy = 8 in Equation (1), we get
(x + y)² = x² + 2xy + y² = x² + 8 + y² = 49. Subtracting 8 from both sides of the last equation, we get
x² + y² = 41.