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Solve each of the quadratics by first completing the square. When the roots are irrational, also give the solutions to the nearest one thousandth.$$x^{2}+6 x+21=0$$

$$-3 \pm 2 \sqrt{3} i$$

Algebra

Chapter 0

Reviewing the Basics

Section 3

Completing the Square

Equations and Inequalities

Oregon State University

Harvey Mudd College

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Idaho State University

Lectures

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Solve each of the quadrati…

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In each of the following, …

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we're solving this quadratic X squared plus six X plus 21 is equal to zero. And since the directions ask us to complete the square, what I'll do is and I know I teach us sometimes a little differently than other people is a just get X squared plus six X and then I put a blank right here is equal to zero minus 21 is negative 21 then another blank. So what goes in that blank? It's always the same process. You look at the B value, which is six in this case, divided by two always divided by two squared Well, three squared is that 65.2 equals nine. So what I'm gonna do is add nine right here. But if I add nine to the left side of the equation, I also have to add nine to the right side of the equation. You might be sitting there and say, Why on earth would you do that? Well, on the left side, this is not a perfect square. China mule. That factors of nine that had to be six R three and three. Always this number. So instead of writing X plus three. Twice we can write at once squared and on the right side. Well, negative 21 plus nine is negative. 12. So why do we do that? Well, now we can solve the square problem by square, rooting both sides. Um, and just as a reminder, when you square root, you get two inches a plus and a minus square root of a negative has to have an eye in there. And I might even rewrite that as the square before times the square root of three because the square root of four is too. We don't know the routes square to three. So we just leave that alone. And so the square root in the square canceled. And there's only one thing left to do, and that's to move this three to the other side. Well, to move a positive term to the other side is to subtract it. And we always write the real part in front of the imaginary Um, and this is a good answer right here.

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