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By completing the square, solve the equation π₯ squared minus two times the square root of three plus one equals zero.
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Before we work on this problem, I wanna talk about completing the square in general.
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If we have an equation in the standard form ππ₯ squared plus ππ₯ plus π equals zero, here are the steps we follow to complete the square.
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First, we divide everything by π.
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We want the coefficient in front of π₯ squared to be one.
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Then we would have the equation π₯ squared plus π over π π₯ plus π over π.
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After that, we need to move this constant, π over π, to the other side of the equation.
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We then have π₯ squared plus π over π π₯ equals negative π over π.
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Hereβs where we start adding things to the equation.
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We need to add π over π divided by two squared to both sides of the equation.
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But we can say divided by two is the same thing as multiplying by one-half, which means weβre adding π over two π squared to both sides of the equation.
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It will now look like this.
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π₯ squared plus π over π π₯ plus π over two π squared equals π over two π squared minus π π.
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The left side of the equation is now a square value.
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It is now π₯ plus π over two π squared.
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And the right side is π over two π squared minus π π.
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You can also simplify this side of the equation.
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If we square π and two π, we get π squared over four π squared minus π π.
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And if we multiply the π over π by four π, in the numerator and the denominator, we can say π squared minus four ππ all over four π squared, which weβll substitute back into this part of the equation.
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Why did we do all this?
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We did all of this because this equation will work no matter what values you have for π, π, and π.
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Just to note, there should be an π₯ by the negative two times the square root of three there.
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The equation weβre solving for is π₯ squared minus two times the square root of three π₯ plus one equals zero.
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And our π value is one.
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The coefficient of π₯ is just one.
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If π equals one, the π value, the coefficient of the π₯ to the first power, is negative two times the square root of three.
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π equals negative two times the square root of three.
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We also need to know what π over two is.
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π over two equals negative two times the square root of three over two, which we can simplify.
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The twos cancel out.
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π over two is equal to the negative square root of three.
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And π over two squared equals the negative square root of three squared, which is three.
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And our π value equals one.
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Remember, we move the π value to the other side of the equation.
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Then we add π over two squared to both sides.
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And our π over two squared equals positive three.
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Negative one plus three equals two.
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The left side is a square.
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And we rewrite that as π₯ plus π over two squared.
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Our π over two is the negative square root of three.
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We can simply say π₯ minus the square root of three squared.
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When our goal is to solve the equation, we want to find out what π₯ values make this statement true.
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And weβll do that by isolating π₯.
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That means weβll need to take the square root of both sides of the equation.
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And that tells us that π₯ minus the square root of three equals the square root of two.
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Here weβre gonna break the equation up into two statements, because we remember that the square root of two has both a positive and a negative solution.
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Weβll have π₯ minus the square root of three equals the positive square root of two.
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And π₯ minus the square root of three equals the negative square root of two.
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To get π₯ by itself, weβll add the square root of three to both sides of both equations.
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π₯ equals the square root of three plus the square root of two.
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Or π₯ equals the square root of three minus the square root of two.