Question 594750: Can you help me solve for x in this equation?
-6x+6x^2=0
I think the first step is to set both sides to 0...
-6x=0, = (-6/-6x = 0/-6), x = 0
and 6x^2=0(But I do not know how to solve this side?
Is this correct? I get lost after this?
Thanks!!
Juanita
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website!
The problem is to solve the equation:
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-6x + 6x^2 = 0
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Before we do anything else, let's arrange the two terms on the left side of the equal sign to be in descending powers of x. It's not necessary to do this, but it's sort of a standard practice. After we make this change the equation is:
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6x^2 - 6x = 0
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(All we have done so far is just switch around the two terms on the left side of the equal sign, and this does not really change the original equation at all.)
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Notice that each of the terms on the left side of the equal sign contains at least one x. Let's factor an x out of each term. When we do that, the equation becomes:
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x(6x - 6) = 0
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(Notice that if we multiply the x times each of the terms in the parentheses, we get back to the above equation of 6x^2 - 6x = 0, so we didn't really change the equation. We just found a different way of writing it.)
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Now the equation has a product on the left side of the equal sign. That product is x times (6x - 6) and it equals the zero on the right side of the equal sign.
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Think about this. If you multiply two things together and the answer equals zero, then one of the multipliers must be a zero. For example, what would you multiply 3 by to make the product equal zero? The answer is 3 times 0 equals 0, so you need to multiply 3 by zero to get zero.
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Compare this simple example to our problem of:
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x(6x - 6) = 0
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In order for us to get the answer of 0 (on the right side) one of the multipliers on the left side must be a zero. This means that either the multiplier x must be a zero or the multiplier (6x - 6) must equal zero in order to get the answer 0. So we can write two equations. Either:
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x = 0
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or
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6x - 6 = 0
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The first equation tells us directly that if x equals zero, this will make the original equation of the problem equal to zero. The second equation needs just a little work to solve for another value of x that will make the original equation equal to zero. So let's solve:
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6x - 6 = 0
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Make the -6 disappear from the left side of the equal sign by adding +6 to both sides:
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6x - 6 + 6 = +6
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On the left side of the equal sign the -6 and the +6 cancel each other and we are left with:
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6x = 6
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Solve for x by dividing both sides by 6 (because 6 is the number of x's on the left side of the equal sign). The division by 6 on the left side results in just x on the left side, and the right side is 6 divided by 6. So
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6x/6 = 6/6
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becomes just
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x = 1
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That's it. We have two values for x that will make the original equation balance and those two values are 0 and 1.
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Let's check to ensure that our answers are correct. Go back to the original equation:
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-6x + 6x^2 = 0
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If we substitute our answer 0 for x, this equation becomes:
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-6*0 + 6*0^2 = 0
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Notice that with zero as a multiplier in each term, the two term on the left side each become zero. So when x equals zero, the original equation becomes:
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0 + 0 = 0
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And that is obviously correct.
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But how about our answer that x equals 1? Again go to the original equation:
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-6x + 6x^2 = 0
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and this time substitute 1 for x to get:
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-6*1 + 6*(1)^2 = 0
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If you multiply out each of the terms on the left side of the equal sign you get:
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-6 + 6 = 0
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And when you add the two terms on the left side the equation becomes:
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0 = 0
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which is correct. So we know that x equals 1 is also a good solution.
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So the two answers to this problem are x = 0 and x = 1 and both of these answers check.
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I hope this explanation helps you to understand this problem. If you have a problem in which x appears in every term, you can factor out an x as we did here. That is something that you can look for.
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