Solve each equation. See Examples 8 and 9. 8(x-4)^4-10(x-4)^2=-3

Question

Accepted Answer

Hey, everyone here we are asked to solve for X in the following equation, 49 times the quantity of X minus one raised to the fourth power minus 260 times the quantity of X minus one squared plus 256 is equal to zero. Now, looking at our given equation, normally, we would want to factor the less side to solve for X. But we see that it would be rather complicated to do so since we have this expression raised to a rather high power, so we want to begin by simplifying our given equation and we can do this by letting the variable U be equal to the quantity of X minus one squared. So now we can just rewrite were given equation as a simplified form. So we have 49 U squared for our first term and then we have minus you plus 256 is equal to zero. So now we see we have a much simpler equation or expression to start factoring. So we can go ahead and begin factoring by first looking at our middle term here, which is negative 260 you. Well, we need to split this middle term into two terms which some to negative 260 you. But the product of their coefficients must be equal to positive 256. So doing this, we will have 49 nine U squared minus 196 U minus 60 for you plus 256 is equal to zero. So now we just want to check our middle terms here. So we have negative 196 U minus 60 for you. And we see simplifying we do in fact Get -260 you. But now we want to check their coefficients. So we have negative 196 times negative 64. And simplifying we do in fact get positive 256. So we see we chose the correct to middle terms so we can go ahead and continue with factory this left side expression. So beginning by looking at our first two terms, we see they have a common factor of 49 you. So factoring out this value, we have 49 you times the quantity of U -4. And then for our last two terms, we can factor out negative 64. So we have minus 64 times the quantity of U minus four is equal to zero. So now we see that both terms have a common factor of this U minus four. So we can go ahead and factor out this Expression and we get the quantity of U -4 times the quantity of 49 U -64 is equal to zero. And now recalling the zero factor property, we know that we can set each individual expression equal to zero to find our values for you. So we can begin with the first part of our expression where we have U -4. And again, we can set this equal to zero. And now solving for you, we need to first add four to both sides and we get U is equal to positive four. And now repeating this with the other part of our expression, we have 49 U -64 is equal to zero. And now solving for you first, we want to just add 64 to both sides. So we have 49 U is equal to 64. And now getting rid of the coefficient, we divide both sides by 49. And we see the other value for you is you is equal to 64 divided by 49. So now our next step here is to replace or substitute back in our original expression for you. So we can start solving for X. So we can begin with the first value we found which is four for you. So we will instead of you, we have the quantity of X minus one squared. Is now equal to four. And now just solving for X, we need to first get rid of the squared term. So we need to take the square root of both sides. But since we are taking the square root of the right side here, we need to add the plus or minus in front of it. So now simplifying, we have X minus one is equal to plus or minus the square root of four, which is the same as plus or minus two. So now we have two different expressions to find value X four. So we first can start with dealing with positive two. So we have X minus one is equal to positive two. And now again, solving for X, we need to add one to both sides and we get one of our, our values for X is just positive three. So now we can work with the negative two expression. So we have X minus one is equal to negative two. And now again, we need to add one to both sides to solve for X. And we see that another value for X is negative one. So now we can repeat this process with the other value of view that we found earlier, which was 64 Divided by 49. So again, substituting in our original expression for you, we have the quantity of X -1 squared is equal to 64 divided by 49. So now again, we need to start by getting rid of the squared value here. So we take the square root of both sides, which means the right hand side gets a plus or minus in front of the value. So simplifying we have X -1 is equal to plus or minus 8/7. So now again, we have two different versions or solutions for X that we can find so we can begin first with the positive eight sevens. So we have X minus one is equal to positive eight sevens. And now solving for X, we just need to again add one to both sides and we get another value for X is 15 7th. And now we can work with the negative value. So we have X minus one is equal to negative 8/ adding one to both sides. We get 1/4 value for X as negative 1/7. So now with all of the values that we found for X, we can rewrite them as a solution set. So we have the value of negative one that we found. We also had negative 1/7 and then we had 7th and we also had positive three. So at this, with our four values we found for X, we have been left with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each equation. See Examples 8 and 9. 8(x-4)^4-10(x-4)^2=-3

Key Concepts

Polynomial Equations

Substitution Method

Quadratic Formula

Watch next