Solve: 2x(x+3)+6(x−3)=−28. (Section 5.7, Example 2)

Question

Accepted Answer

Welcome back. I am so glad you're here. We are asked to solve the following equation. Our equation is three X multiplied by the sum of X plus five plus 15, multiplied by the difference of X minus five is all equal to negative 123. Our answer choices are answer choice A X equals negative 55. Answer choice B X equals negative 10, negative six. Answer choice C X equals negative eight, negative two and answer choice D no solution. All right. In order to solve for X, we want to get all of our XS together. First, we see that we have some XS outside of parentheses and some inside. So let's distribute what's in front of the parentheses. We'll distribute the three X to the X and the five and the 15 to the X and the negative five. In order to break those XS free of those parentheses. So three X multiplied by X is three, X squared. Three X multiplied by positive five is plus 15, X, 15, multiplied by X is again plus 15, X and 15 multiplied by negative five is negative 75. That's all still equal to negative 123 we can combine the like terms. Here in the middle, we have 15 X plus 15 X which will give us 30 X. So we'll have three X squared plus X minus 75 equals negative 123. And we notice that here, we have a quadratic, we have an X squared and then an X to the first power and a constant term. And what we want to do when we have a quadratic is set one side equal to zero. And see if we can factor. So we want the right hand side equal to zero. Here, we have a negative 123 over there. So we'll add 123 to both sides of the equation. And on the right negative 123 plus 123 is zero. So that is what we want on the left, we bring down three X squared plus 30 X and then negative 75 plus 123 is a positive 48. Now, we can do the reverse of the foil process to see if we can factor this quadratic. We set up two open sets of parentheses equal to zero. And noticing here that we've got something in front of our X squared. There's a three there. So we want to see if we can get rid of that three first. Can we factor three out of all of these terms? And yes, we can. So what we can do before we start factoring is divide everything by three. And that'll make it easier to factor that out with the reverse boil process not divided by zero. That would be undefined divided by three. So three X squared divided by three is X squared. 30 X divided by three is plus 10, X 48 divided by three is plus 16. And that's all still equal to zero because zero, divided by three is zero. Now, attempting to factor will be easier. We ask ourselves what multiplied by what equals X squared and that will be X multiplied by X, then what multiplied by what gives us a positive 16? Keeping in mind that when we multiply our outsides and add those to our insides, they need to add up to a positive 10 X. So in other words, what two factors of 16, add up to a positive 10, that would be a positive eight and a positive two. And now we know that when these two factors are multiplied and they equal zero, that either one of them could equal zero for this to be true. So we set each of these factors X plus eight and X plus two equal to zero to solve for X with X plus eight equals zero. We subtract eight from both sides on the left eight minus eight is zero. So we have just X on the left and that's equal to, on the right, zero minus eight is negative eight. Now, for X plus two equals zero, again, we subtract two from both sides on the left two minus two is zero and all we have is X and on the right zero minus two is a negative two. And we have solved for X X is equal to negative eight and negative two. We look at our answer choices and that matches with answer choice. C well done. We'll catch you on the next one.

Solve: 2x(x+3)+6(x−3)=−28. (Section 5.7, Example 2)

Key Concepts

Distributive Property

Combining Like Terms

Solving Linear Equations

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