AMC 10 · 2013 · #11
Grade 8 algebraPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
One equation with two unknowns usually cannot pin down each value, so a plain solve will not work. Instead, rearrange the equation and complete the square to rewrite it as a sum of squares. That new form is more useful: a sum of squares can only equal 0 when every square is 0, which forces exact values for x and y.
Move all terms to one side
Subtract 10x, add 6y, and add 34 on both sides so the right side becomes 0.
Getting 0 on one side sets up a form you can rewrite and read off.
6.EE.A.3Organize Information In More WaysComplete the square for each variable
Half of 10 is 5 and half of 6 is 3, so x² - 10x becomes (x - 5)² - 25 and y² + 6y becomes (y + 3)² - 9.
Completing the square turns a scattered quadratic into a clean square plus a leftover number.
6.EE.A.3Organize Information In More WaysCombine the leftover numbers
Substituting both pieces back, the constants -25, -9, +34 cancel, leaving (x - 5)² + (y + 3)² = 0.
-25 - 9 + 34 cancels to 0, leaving a bare sum of two squares.
7.NS.A.1Organize Information In More WaysForce each square to 0
Two squares sum to 0 only if both are 0, so x = 5 and y = -3, giving x + y = 2, choice (B).
Two things that can never be negative can only sum to 0 by both being 0.
Two things that can never be negative can only add to zero by both being zero.
▸ Why?
A square is never negative, so each term sits at or above zero.
▸ Why?
If one were positive the other would have to be negative to cancel it, which is impossible.
Complete the square to turn the equation into a sum of squares equal to 0, then remember a sum of squares is 0 only when every square is 0.
- Move all terms to one side
- Complete the square for each variable
- Combine the leftover numbers
- Force each square to 0