AMC 10 · 2002 · #1
Grade 8 arithmeticPick an answer.
Instead of multiplying everything out, look at the shape of the left side: it is (2x+3) times one thing plus (2x+3) times another thing. Tool #16 (Change Focus) says read that structure rather than expanding, because the shared factor (2x+3) can be pulled out. Once it is factored into a product that equals zero, tool #7 (Identify Subproblems) takes over: a product is zero exactly when one of its factors is zero, so the single hard equation splits into two easy linear equations. Solve each for one root, then add the roots.
Pull out the shared factor
Both terms share (2x+3); pull it out and the bracket becomes 2x-10, leaving (2x+3)(2x-10)=0.
When every piece carries the same factor, you can grab that factor once instead of multiplying it back in.
6.EE.A.3Change Focus Count The ComplementSplit into two linear equations
A product is zero only if a factor is: 2x+3=0 gives x=-3/2, and 2x-10=0 gives x=5.
If two numbers multiply to zero, one of them had to be zero, so each factor hands you one root.
Because the factored expression equals zero, each factor gives one root on its own.
▸ Why?
If two numbers multiply to zero then at least one of them was already zero, so the equation splits into one simple case per factor.
▸ Why?
The shared factor could be lifted out of the whole expression in the first place because a common factor divides every term alike.
Add the two roots
Add the two roots over a common denominator: -3/2+10/2 = 7/2, choice (A).
Adding a negative fraction to a whole number is just lining them up over the same denominator.
7.NS.A.1Identify SubproblemsWhen two terms share a factor, pull it out first — a product that equals zero breaks the problem into one easy equation per factor.
- Pull out the shared factor
- Split into two linear equations
- Add the two roots