AMC 10 · 2013 · #6
Grade 8 algebraPick an answer.
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
Everything moves to one side.
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
Each variable completes into a square.
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
The constants cancel to exactly zero.
-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
Both squares vanish, giving 2, choice (E).
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?
If either one were positive the total would be positive, since the other can never pull it back down.
▸ Why?
A total of zero means nothing is left over, so neither square can contribute anything at all.
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