COMC Number Theory and Algebra Flashcards

If x + y = 5 and xy = 6, what is the value of x² + y²?

Answer: 13

Using x² + y² = (x+y)² − 2xy = 25 − 12 = 13.

For what value of k does kx² − 6x + 3 = 0 have exactly one real solution?

Answer: 3

Equal roots require discriminant = 0: (−6)² − 4(k)(3) = 36 − 12k = 0, giving k = 3.

If the roots of x² − px + q = 0 are 3 and 5, what is p² − 4q?

Answer: 4

By Vieta's formulas p = 8 and q = 15, so p² − 4q = 64 − 60 = 4.

What is the maximum value of the expression 6x − x² − 5?

Answer: 4

Completing the square gives −(x−3)² + 4, so the maximum value is 4 at x = 3.

If a/b = 3/4 and a + b = 28, what is the value of a − b?

Answer: −4

Setting a = 3k and b = 4k gives 7k = 28, so k = 4, a = 12, b = 16, and a − b = −4.

What is the value of (1 + √3)(1 − √3)?

Answer: −2

By the difference of squares identity: (1+√3)(1−√3) = 1² − (√3)² = 1 − 3 = −2.