Disclaimer: Not “solved”. Simplified!
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One of my grad school professors called the mistake (a+b)^2 = a^2 + b^2 the “freshman dream”. I guess he meant freshmen in college, but freshmen in high school make the same mistake. Come to think of it, a lot of people make the same mistake. What other mistakes are common in Algebra?
Here are just a few I can think of:
x = 1. NO. Well, sometimes, but definitely not always.
-7 + 5 = -12. NO. This one’s a never and the reason I developed the ZeroSum ruler.
(a+b)^2 = a^2 + b^2. NOPE. There’s a middle term in there. Write it out. Don’t be lazy. Find it.
1/2 = 2. NO. “1 out of 2″ is not 2.
(x + 2) + (x - 5) = x^2 – 3x - 10. NO. This is an addition problem, not a multiplication problem.
(x+2) – (x - 5) = -3. NOPE again. That little – sign is a -1 in disguise and needs to be distributed.
(2/5)(7/3) = ….<blank>…. NO. Multiply the numerators together and the denominators together to get 14/15. Easy!
“Thirty percent of 140 = 140/.3″. NO. You’re going to get a huge number. “OF” means to multiply, not divide.
|-5| = 5, so |5| = -5. NO. Absolute value is always a positive number. It represents a distance. Even if you walk backwards, you’re still moving some distance.
“find x^2 if x = -5″. Answer: -25. NO. Jut because the TI-83 says -25 doesn’t mean it’s right. Calculators use PEMDAS, which states multiplication comes after exponents. Remember a few mistakes back that the little – sign is a -1 in disguise. It’s the same here. If you put “-5^2″ into the calculator, the calculator will square 5 and then multiply by -1. To square the entire -5, use ( ). Or just remember that a negative number raised to any even power will always be positive.
(x^4)(x^5) = x^20. NO! This is a common mistake kids make. But if they start by thinking about what “x^4″and “x^5″ mean, it’s easy to see that (x^4)(x^5) expands to (xxxx)(xxxxx). Parenthesis right next to each other tell us to multiply, so you end up with (xxxxxxxxx) or x^9 or x^(4+5).
(x^4)^5 = x^9. No, too. x^4 = (xxxx) and the “^5″ means you have 5 (xxxx)’s. So (x^4)^5 expands to (xxxx)(xxxx)(xxxx)(xxxx)(xxxx). Parenthesis right next to each other still say “multiply” so we have 20 x’s in a row, or x^20.
I know there are more. What did I miss?