Solve each equation in Exercises 41–60 by making an appropriate s...

Question

Solve each equation in Exercises 41–60 by making an appropriate substitution.

(x - 5)^2 - 4(x - 5) - 21 = 0

Accepted Answer

Welcome back. I'm so glad you're here. We've got this equation parentheses X minus two squared minus eight times parentheses X minus two. And parentheses minus nine equals zero. And we get to solve this. Using the substitution method. What is the substitution method? When you've got an equation that isn't straightforwardly factory ble. But you notice that the factors or the variables are kind of similar with the first two terms and one of them has an exponent. That is double the other one because this one is an invisible one here. Then you can take that factor or that variable and you can set it equal to whatever variable you want. I'm going to use W and then substitute a W. N. To your equation. So no longer x minus two squared. But now W squared minus eight times no longer x minus two. But w -9 equals zero. And hey, that looks much more factory bubble. So now let's give that a try. We set our parentheses and what times what gives us W squared. Well that'll be a W times W. And what times what gives us negative nine. Well we'll have to have a positive and a negative number. But we've got options with threes and one in nine. So we look to the middle and we say we see that they have to add up to equal negative eight. So in this case let's go with a negative nine and a positive one that's still equal to zero. And we can always foil this to check to make sure we did this correctly. So let's do that very quickly. Our first W times W W squared our Outsides W times one is plus one. W insides negative nine times W is negative nine W. And r lasts negative nine. Times one is negative nine, combine our middle like terms W squared one, W minus nine, W is minus eight, W minus nine equals zero. And yeah, this matches. We chose the correct factors. So now we proceed to set those two factors, W minus nine, N. W plus one equal to zero. So we'll go ahead and add nine to both sides. For W minus nine equals zero and we get W equals nine and for our other one, W plus one equals zero will subtract one from both sides and we get W equals negative one. And yeah, but before you go, congratulating yourself and clicking an answer here, we need to recall that W equals x minus two and we're actually trying to solve for X. So anywhere we have a W we are going to be substituting a back in our x minus two. So instead of W equals nine it's x minus two equals nine and instead of W equals negative one, it's x minus two equals negative one. But now we can solve for X and x minus two equals nine will add two. Table sides and we get X equals and an x minus two equals negative one. We will add two to both sides again and we will have X equals one. We look at our answer choices and that matches with answer choice a well done, we'll catch you on the next one.

Solve each equation in Exercises 41–60 by making an appropriate substitution. (x - 5)^2 - 4(x - 5) - 21 = 0

Key Concepts

Substitution Method

Quadratic Equations

Factoring

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