You can put this solution on YOUR website! Given:
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Solve this by completing the square and then by factoring.
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Let's use the factoring method first. It is so easy, and we can use the answers we get to
check the results we get from completing the square.
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In the factoring method, you can see that y is a factor of both terms on the left side.
Factor a y and the given equation becomes:
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Now notice that this equation will be true if either of the factors equals zero because
a multiplication by zero on the left side will make the left side be zero and therefore
it will equal the right side.
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So we set each of the factors equal to zero ...
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First y = 0. No solving is necessary. That is one answer for y.
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Next y - 5 = 0. If you add 5 to both sides this becomes y = 5. And this is the second
answer for y.
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Now let's try completing the square.
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Start with
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Take half of the multiplier of y. The multiplier is -5 and half of that is .
Square this to get . This is the number that you need to add to both sides
of the equation to make the left side a perfect square. When you add to both
sides of the equation you get:
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The left side of this equation is a perfect square. The quantity that squares to this left
side is:
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and the equation then becomes:
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Now take the square root of both sides and recognize that the right side will have two
possibilities ... one with a plus sign and one with a minus sign. Therefore, after taking
the square root of both sides you get:
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You can solve for y in both of these equations by adding to both sides. In
the first of these when you add to both sides then:
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becomes
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and the other equation ... when you add + to both sides
becomes because adding to both sides cancels the on
both sides.
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These two answers agree with the answers that we got by factoring, but factoring is a
much quicker way to do the problem with less chance for making an error.
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Hope that this helps you to understand the problem and to recognize that if you can factor
the left side, it is generally easier than completing the square or using the quadratic
formula for that matter.
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