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Solve the equation negative 𝑥 squared plus seven 𝑥 plus one equals zero.
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We can start by copying down the equation exactly as it was listed in the problem.
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And then I noticed this negative leading coefficient.
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Since I’m going to have to do some factoring or may be completing the square, I want to make sure that my leading coefficient is positive, and I can do this by moving it to the other side of the equation.
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I add 𝑥 squared to both sides.
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But I don’t only want to move my leading coefficient; I wanna keep the whole equation on the same side.
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This means that I’ll subtract seven from both sides of the equation as well.
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I’ll also subtract one from both sides of the equation.
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Essentially, we’ve just flipped the problem.
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We now have zero equals 𝑥 squared minus seven 𝑥 minus one.
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I’m starting to think about how to factor and solve this equation, and I looked to the third term.
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If I wanted to factor the problem like this, I would need to take two factors of the third term, something like this, 𝑥 minus one 𝑥 plus one.
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The problem is, we have this third term of negative seven, so this kind of factoring won’t help us solve this problem.
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To solve this problem, we’ll have to use a strategy called completing the square.
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The first step here would be to move the whole number one back to the other side of the equation.
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We’re left with one equals 𝑥 squared minus seven 𝑥.
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Completing the square works by taking the coefficient of the middle term, the b here.
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We then use whatever that b value is, and we divide it by two and then we square it.
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After we do that, we add that value to both sides of the equation.
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In our case, the b equals negative seven, so we’ll need to add negative seven over two squared to both sides of our equation.
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That would be negative seven squared over two squared, which simplifies to 49 over four.
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We need to add 49 over four to both sides of our equation.
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Notice that, instead of saying 49 over four plus one, I said 49 over four plus four over four.
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Four over four is equal to one, and we can go ahead and add these terms together if we give them a common denominator.
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The rest we just copy down.
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49 fourths plus four-fourths equals 53 fourths.
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Now we need to factor 𝑥 squared minus seven 𝑥 plus 49 over four.
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And if you didn’t immediately recognize the pattern, that’s okay, because completing the square tells us that the factor of this problem will be equal to 𝑥 plus b over two squared.
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We add 𝑥 plus; remember our b was negative seven over two squared, but we still haven’t solved the equation.
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Solving the equation means we know what 𝑥 equals.
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Now we’ll need to get 𝑥 completely by itself.
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To get rid of this square, we’ll need to take the square root of both sides of our equation.
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Now we’re left with the square root of 53 over the square root of four equals 𝑥 plus negative seven over two.
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From this line, we can take the square root of four, which equals two.
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We’ll keep the square root of 53 and say that we need the positive square root of 53 and the negative square root of 53.
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To get 𝑥 by itself, we’ll need to add seven halves to both sides of the equation.
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On the right side, negative seven halves and positive seven halves cancel out.
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On the left side, we have seven halves plus or minus the square root of 53 over two.
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That plus or minus there tells us that there will be two solutions here.
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The first solution will be seven plus the square root of 53 over two and the second solution would be seven minus the square root of 53 over two.
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These are the two solutions of the equation negative 𝑥 squared plus seven 𝑥 plus one.