Solve each equation in Exercises 1 - 14 by factoring.

10x - 1 = (2x + 1)^2

Question

Solve each equation in Exercises 1 - 14 by factoring.

10x - 1 = (2x + 1)^2

Accepted Answer

Hello, everyone. In this video, we're gonna be looking at this practice problem where we're trying to solve for value of X in the equation 32 X plus 41 is equal to four X minus three squared. And we want to try to solve by factoring. So in this case, you can see that the left-hand side is fully expanded, but my right hand side is four X minus three squared. So I'm gonna go ahead and expand that exponent to be four X minus three times four X minus three. And I'll multiply those two terms using the foil method. So I have the first terms being multiplied four X times four X is X squared. And the outer terms being multiplied four X times negative three, negative 12 X. Then the inner terms negative three times four, X is negative 12 X and negative three times negative three is positive nine. And notice I can combine the middle terms negative 12 X minus X. So that leaves me with 16 X squared minus 24 X plus nine. And our left hand side is still 32 X plus 41. So in this case I want to try to solve by factoring. And in order to do that, I need to make sure that I have all my terms on one side. So I can then factor that one equation. So I'm gonna do that by subtracting by 41 and S. And if I do that, my left hand side is now zero and my right hand side is 16 X squared minus 56 X minus 32. So now I have this new equation that I can factor and notice how all of these numbers are even 16 56 and 32. So I can divide them by two. But if I think a little bit about a bigger number, I can divide them by notice that eight is also a common factor between all of them. So if I take out 8 16, divided by eight is two. So I'm left with two X squared and 56 divided by eight is seven. So some I left with negative seven X and 32 divided by eight is four. So I'm left with negative four. Notice how my left hand side is still zero. And if I divide by eight on both sides, the eight on my right hand side cancels and I'm still left with zero on my left hand side because zero divided by eight is still zero. So now I have zero is equal to two X squared minus seven X minus four. And now I can factor out the equation two X squared minus seven X minus four. Well, in this case, notice how two X squared minus seven X minus four follows the form for a quadratic equation A X squared plus BX plus C. And recall that one where factoring a quadratic equation, we want the two terms to multiply to get AC and the two terms to add to get B. So in this case, I'm gonna use this rubric to try to help me find the factor terms. So in our problem, ac is equal to two times negative four or negative eight and B is equal to negative seven. So I'm trying to find factors of negative eight that will add to negative seven. So if I try two and negative four multiply together, I get negative eight but added together I get negative two. And if I switch the sign multiply together, I still get negative eight and added together I get positive two. So I know that those factors are not it. And if I try negative one and eight multiply together, I get negative eight and added together I get positive seven. So I need to switch the sign between these two and now multiplied together negative eight and positive one. I get negative eight and add it together negative eight plus one. I get negative seven. So I know that these are the factors that I care about or that I'm going to use to help me factor these two. So in this case, I'm going to replace my middle term with these two factors. So my equation is going to become two X squared plus one X minus eight X minus four. And in this case, I'm gonna factor these terms by sort of grouping them together and I'm gonna group them based on this ratio where I have four divided by eight is equal to one half. And I have one divided by two is also equal to one half. So I'm gonna make two X squared plus one X my first grouping and I'll make negative A X minus four, my second grouping. So if I look at my first grouping, two X squared plus X, I can factor out an X. And if I do that, I have X on the outside multiplied by two X plus one. And then I can look at my second grouping and I can take out a negative four and I'm left with two X minus one, sorry plus one because we took out negative four and negative four divided by negative four is positive one. So now notice how I have two X plus one and two X plus one on both of these terms. So I'm gonna factor two X plus one out. If I do that, I have two X plus one and I'm left with X minus four. So now I have these two terms that I factored out for my equation. And I'm gonna use these two terms to solve for values of X. So first, I have two X plus one equal to zero and I sulfur X I subtract by one and off with two, X is equal to negative one, dividing by two. I have X is equal to negative one half. And my second term X minus four is equal to zero. If I add four to both sides, I have X is equal to four. So now I have X is equal to four and negative one half as my two solutions for when I factor out this equation. So you can see that, that aligns with answer choice A. So hopefully this video helped and see you next time.

Solve each equation in Exercises 1 - 14 by factoring. 10x - 1 = (2x + 1)^2

Verified Solution

Key Concepts

Factoring

Quadratic Equations

Zero Product Property