```Question 622630
Given to solve by factoring:
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{{{x^2 +(x + 2)^2 = 650}}}
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First, square the terms in the parentheses by multiplying:
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{{{(x+2)*(x+2)}}}
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When you do this multiplication you get:
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{{{x^2 + 4x + 4}}}
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Substitute this into the original problem in place of {{{(x+2)^2}}} and the problem then becomes:
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{{{x^2 + x^2 + 4x + 4 = 650}}}
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Combine the two x-squared terms to get:
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{{{2x^2 + 4x + 4 = 650}}}
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Subtract 650 from both sides and you have:
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{{{2x^2 + 4x -646 = 0}}}
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You can simplify this a little by dividing both sides (all terms) by 2 to get:
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{{{x^2 + 2x - 323 = 0}}}
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Now you can do the factoring process.  The negative sign on the 323 tells you that you will have one positive factor and one negative factor whose product gives you negative 323. The coefficient of the x term is +2. This tells you that the positive factor must be bigger than the negative factor by 2.
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Now you can make some educated guesses. If the two factors were +20 and -18, their product would be (you can do this in your head) -360. You need two numbers whose product is -323, so you are pretty close, but the factors you guessed are a little too big because they give a product of -360. How about trying +18 and -16.  The product of these two numbers is -288 and since this is smaller that -323, you need two bigger numbers. If you now try +19 and -17, you will find that their product is -323, just as you need it to be. So you know that the product of the two factors is:
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{{{(x+19)*(x - 17))}}}
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and if you set these equal to zero as above you have:
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{{{(x+19)*(x-17)=0}}}
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Notice that this equation will be true if either of the factors equals zero. So you can set both factors equal to zero and solve for x to get:
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{{{x + 19 = 0}}}
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and solve for x to get x = -19
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Then:
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{{{x - 17 = 0}}}
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and solve for x to get x = +17
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Those are the two solutions for x in the equation:
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{{{x^2 + (x+2)^2 = 650}}}
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You can check the solutions by first substituting -19 for x and you have:
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{{{(-19)^2 +(-19+2)^2 = 650}}}
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When you square -19 you get +361. and when you add -19 and +2 in the parentheses you get -17. Then square that and you have get +289. Substituting these values into the equation gives:
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{{{361 + 289 = 650}}}
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And if you add the terms on the left side you find that the equation balances, so x = -19 works.
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You can do the same kind of analysis for the second answer. Go back to the original equation and substitute +17 for x and you find that it also checks out.
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Hope this helps you see where you went slightly off track and lets you see how the answers can be found.
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