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Find the roots of the quadratic equation 𝑥 plus four all squared plus eight equals zero.
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Well, we’ve got our quadratic equation, which is 𝑥 plus four all squared plus eight equals zero.
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Well, we might be tempted here to distribute across our parentheses, but in fact, it’s much more simpler than that.
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Well, if we subtract eight from each side of the equation, what we’re gonna get is 𝑥 plus four all squared equals negative eight.
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Okay, so now what we can do is take the square root of both sides of our equation.
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And when we do that, what we’re gonna get is 𝑥 plus four is equal to positive or negative the square root of negative eight.
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Now it’s at this point that we might think, well, hold on, there aren’t any solutions.
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Because if we look at the square root, we’ve got the square root of a negative number.
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However, there are roots if we look at complex roots.
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So let’s take a look at root negative eight.
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Well, if we apply our surd or radical rule, we can rewrite root negative eight as root eight multiplied by root negative one.
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And what we know is that root negative one is equal to the imaginary number 𝑖.
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So therefore, what we can do is rewrite our root negative eight as root eight 𝑖.
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However, because we’ve got root eight, we know that we can simplify this a little bit more.
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And that’s because we can split root eight into root four root two.
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So we’ve got root four root two 𝑖.
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So therefore, we can rewrite root negative eight as two root two 𝑖.
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Okay, so let’s put this back into our equation.
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When we do, what we get is 𝑥 plus four equals positive or negative two root two 𝑖.
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So then, if we subtract four from each side of the equation, we get 𝑥 is equal to negative four plus or minus two root two 𝑖.
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So therefore, we can say that the complex roots of the quadratic equation 𝑥 plus four all squared plus eight equals zero are negative four minus two root two 𝑖 and negative four plus two root two 𝑖.