If x + y = 10 and xy = 21, what is the value of x^2 + y^2? (2021)

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Question: If x + y = 10 and xy = 21, what is the value of x^2 + y^2? (2021)

Options:

Correct Answer: 49

Exam Year: 2021

Solution:

We know that x^2 + y^2 = (x + y)^2 - 2xy. Substituting the values, we get (10)^2 - 2(21) = 100 - 42 = 58.

If x + y = 10 and xy = 21, what is the value of x^2 + y^2? (2021)

If x + y = 10 and xy = 21, what is the value of x^2 + y^2? (2021)

Step 1: Start with the equations given: x + y = 10 and xy = 21.
Step 2: Recall the formula for x^2 + y^2, which is x^2 + y^2 = (x + y)^2 - 2xy.
Step 3: Substitute the value of x + y into the formula: (10)^2.
Step 4: Calculate (10)^2, which equals 100.
Step 5: Now substitute the value of xy into the formula: 2(21).
Step 6: Calculate 2(21), which equals 42.
Step 7: Now, subtract the result from Step 6 from the result from Step 4: 100 - 42.
Step 8: Calculate 100 - 42, which equals 58.
Step 9: Therefore, the value of x^2 + y^2 is 58.

Algebraic Identities – The question tests the understanding of the identity x^2 + y^2 = (x + y)^2 - 2xy, which relates the sum and product of two variables to their squares.
Systems of Equations – The problem involves solving a system of equations to find the values of x and y indirectly through their relationships.