Question 405235: Factor the expression
49x^2-14x+1
Answer by IWork4Dessert(60)   (Show Source):
You can put this solution on YOUR website! There are multiple ways to solve an expression like this. I'll just explain them all and you can choose which you like best.
1. The "Big X"
This is the most generic way to factor an equation. It will work pretty much every time with a standard factoring equation. Just draw an X that's about an inch tall on your paper. Then pull the middle term from the expression(-14x) and write it in the bottom opening in the X.
Next multiply the first term(49x^2) by the last term(1) and write the result in the top box.
Then it takes a bit of practice. You need to find two numbers that, when multiplied together equal the top number, but at the same time you can add them together to find the bottom number.
Example:
x^2-13x+30
Write -13x in the bottom and 30x^2 in the top. Now find two numbers where, when multiplied together, they equal 30x^2, but when you add them also equal -13x. The numbers are -10x and -3x. Write one in the left indentation in the X, one in the right.
Go back to your original equation and copy the first term down on the next line(x^2). Next write the term in the left indentation of the X(-10x) and then the term in the right indentation in the X(-3x). Put down the last term of the original equation last(30).
x^2-10x-3x+30
Now you want to find the GCF of the first two numbers and the last two numbers. Group them together.
x^2-10x
becomes
x(x-10)
and
-3x+30
becomes
-3(x-10)
Make sure that the terms in parentheses are the same. That's very important. Find a GCF for the second term that makes the part in parentheses the same as the first.
Now take the GCFs of both of them(x and -3) and put them together in parentheses next to the terms already in parentheses.
(x-3)(x-10)
That's your final factored answer.
Now that I'm done shpealing about that whole method, here's how you do your problem.
a) 49x^2-14x+1: The -14x goes in the bottom, 49x^2 in the top.
b) The two numbers on either side must be -7x and -7x(when multiplied together they make 49x^2, when added together they make -14x).
c) Now substitue those numbers for the middle term. 49x^2-7x-7x+1
d) Find the GCF of the first equation and pull it out. 7x(7x-1)
e) Find the GCF of the second equation that gives the same answer in parentheses. -1(7x-1)
f) Record your GCFs together and your terms in parentheses together. (7x-1)(7x-1)
g) Since the two answers in parentheses are exactly the same, you know that it's a "difference of squares". This means that you can just write your answer as (7x-1)^2.
That's that method.
Now to really tease your brain, here's a whole different, whole lot faster way.
2. The Squares
If you see an equation like yours, where the first term(49x^2) and the last term(1) are both squares, you can use the square method.
So instead of doing anything else, just find the square root of the first term(7x) and the second term(1) and put them together. HOWEVER. Since the middle term(-14x) is negative, you have to take that sign and stick it in the middle of the equation. So instead of having (7x+1)^2, you would have (7x-1)^2.
There's your answer, (7x-1)^2.
I would explain the difference of squares method, but you likely don't need to know that and have had enough of my remarkably long explanations anyway.
3. Checking your answers
For the "big X" method, multiply every term by each other. Use rainbows: Multiply the first term of one set by the first term of another, the first term by the second term, the second term by the first term, the second term by the second term, everything. Make sure that your answer is the exact same thing as your original expression.
For the square method, add the two terms together and multiply the answer by two. If it equals the middle term of your original expression, it's correct.
Phew! I need to go give my fingers a break.
I hope the past half an hour of typing helps you out!
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