Question Video: Solving Quadratic Equations in Completed Square Form Mathematics

Find the roots of the quadratic equation (𝑥 + 4)² + 8 = 0 .

02:07

Video Transcript

Find the roots of the quadratic equation 𝑥 plus four all squared plus eight equals zero.

Well, we’ve got our quadratic equation, which is 𝑥 plus four all squared plus eight equals zero. Well, we might be tempted here to distribute across our parentheses, but in fact, it’s much more simpler than that. 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. Okay, so now what we can do is take the square root of both sides of our equation. 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.

Now it’s at this point that we might think, well, hold on, there aren’t any solutions. Because if we look at the square root, we’ve got the square root of a negative number. However, there are roots if we look at complex roots. So let’s take a look at root negative eight. Well, if we apply our surd or radical rule, we can rewrite root negative eight as root eight multiplied by root negative one. And what we know is that root negative one is equal to the imaginary number 𝑖. So therefore, what we can do is rewrite our root negative eight as root eight 𝑖.

However, because we’ve got root eight, we know that we can simplify this a little bit more. And that’s because we can split root eight into root four root two. So we’ve got root four root two 𝑖. So therefore, we can rewrite root negative eight as two root two 𝑖. Okay, so let’s put this back into our equation. When we do, what we get is 𝑥 plus four equals positive or negative two root two 𝑖. 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.

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.

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