Solve each equation. 2x⁴-7x²+5=0

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Accepted Answer

Hey, everyone here we are asked to solve for X in the following equation where we have nine X raised to the fourth power minus 328 X squared plus is equal to zero. Now, normally we would want to factor the left side of our equation to solve for X. But here we see, we have the highest power bi X to the fourth power. And while we're used to factoring expressions with the highest power being too, so we want to begin by simplifying our equation and we can do this by letting the variable U be equal to X squared. So with this statement, we can rewrite our given equation as a simpler form. So we will have nine U squared -328 U plus 144 is equal to zero. So now with our rewritten equation, we see we have a much simpler expression to work with and factor. So we can go ahead and begin factory. And now we want to look at our middle term here and we want to rewrite this middle term as two terms whose sum is still equal to this middle term of negative 328 you. So doing this, we will have nine U squared and then we will have the middle term split up as minus for you and minus 320 for you. And then we still have plus 144 is equal to zero. So now we can start factoring this expression and we can start by looking at our first two terms. And we see here that we have a common factor of you. So we can factor out you and we have you times the quantity of nine U minus four. And then for our last two terms, we can factor out negative 36. So we have minus 36 times the quantity of nine U minus four is equal to zero. So now here with our rewritten expression, we see we now have a common factor of nine U minus four. So we can now factor out this expression here and we get our rewritten equation as the quantity of nine U minus four times the quantity of U minus 36 is equal to zero. Now, recalling the zero factor property, we know we can set each individual equation equal to zero to find our values for you. So we can stop, start with the first part of our expression where we have nine U minus four. And again, we can set this equal to zero. And now solving for you. First, we need to add four to both sides of the equation and we get nine U is equal to four. Now, we need to get rid of the coefficient. So we divide both sides by nine. And we see that one of our values for you is 4/9. And now we can repeat this with the other part of our expression you -36. And again, we can set this equal to zero to solve for you. Now, we need to add 36 to both sides to get you alone. So we see that the other value for you is 36. And now originally we're told to solve for X. So our next step is to substitute in our original expression that we had for you into our two equations that we found. So we can start with our first equation where we have U is equal to 49. And well, again, we need to substitute back in X squared. So we have X squared is equal to 4/9. And now we need to get X alone by getting rid of the squared value. So to do this, we need to take the square root of both sides. But since we are taking the square root of 4/9 we need to add a plus or minus in front of the square root. So now simplifying, we see that X is equal to plus or minus two thirds. So here we have just found two values for X, both positive two thirds and negative two thirds. So now we can repeat this for our other solution that we had for you. So we had U is equal to 36. And again, we substitute back in X squared. So now we have X squared is equal to 36. And again, we need to get X X to be alone by getting rid of the square. So we need to take the square root of both sides. And since we are putting a square root over 36 we need to have a plus or minus in front of it. So simplifying, we have X is equal to plus or minus six. So here we found two more values for X which are positive six and neg six. So in total, we have found four solutions for X, of which we can write as a solution set. So we have found negative six, negative two thirds, positive two thirds and positive six. And so we have found four values for X which leave us with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

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