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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.$$(2 x-3)(3 x+5)=-20$$

$$\frac{-1 \pm \sqrt{119} i}{12}$$

Algebra

Chapter 0

Reviewing the Basics

Section 3

Completing the Square

Equations and Inequalities

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Lectures

02:00

Solve each of the quadrati…

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04:01

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01:45

In each of the following, …

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there's a couple ways to complete the square in this problem. I'm just going to show you what I would do, uh, with this quadratic and the first thing I would do is a distribute. So this problem actually looks doable. Um, so yeah, it's my first step. Distribute my two X in there. So we're looking at six x squared plus 10 X just simple multiplication. And then distribute the negative three and there's minus nine X minus 15, uh, still equals negative 20 on the right side. But now what I need to do actually need to do two things at the first thing I'm gonna do is just combine like, terms. I'm gonna put a blank right here, but I mean, do one more seven, and I need to add this 15 over. That's my one more step. So it cancels out, and I like to leave this as a negative five and again blank. But before I fill in that blank what I actually need to complete the square is this term to be one. So what I tend to do, There's a couple options. You could divide everything by six and you that would work other teachers might say only divide this piece by six. Um, and that way you're not changing the value of that side. But either way, you go about doing this. So I'm just going to commit to this process and continue. You always take half of this number. So half of or divided by two is the same thing is multiplied by one half and then square that number. So we're looking at 1/12 being squared. And when you square one, you get one. And when you square 12, you get 144. But technically, I'm not adding 144 to the left side. I'm adding 6/1 44 uh, and six goes into 144 24 times. So what I'm actually adding is 1/24 to the left side. Because if I distribute, that's a value. So if I add a number to the left side, I need to add the same number to the right side. The reason why you do that is now you can factor the left side, and it's always this number of factors. About one over 144 that had to be 16 It's always that half of 16 number that goes right here and let's see, getting the same denominator five is the same thing as 120 over 24. I can double check my math on that. If you divide, you'll get five. Um, but the reason why I'm doing that is then I can add with the same denominator and they give 120 plus one is negative 119 so that my next step is to get rid of this six. And that's what I meant earlier, as you could have multiple divided the six at the beginning. A lot of teachers do it at this stage. Um, and we actually already did problems similar to this because, uh, six times 24 is 144. Um, let me just rewrite that left side, because then my next step is to undo the square by square rooting square root everything Square root cancels out the square. So on the left side, you're left with X plus 1/12 and on the right side, we can't square root of negative unless we write plus or minus I, and 119 does not break down to any perfect squares. No perfect squares go into it, and they scored. 144 is 12. Um, so we have just one more thing to do, and that's that. Subtract 1/12 over and because it's the same denominator can actually combine them into a single fraction. That's how we add or subtract fractions as the denominators has to be the same, and this is it is your final answer.

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