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Question 345420: I have to solve the following by factoring
9x^3/2-37x^1/2+4x^-1/2=0
Found 2 solutions by haileytucki, MathTherapy:
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! (9x^(3))/(2)-(37x)/(2)+(4x^(-1))/(2)=0 Any \ signs stand for +- and the ~ sign stands for the square rot symbol.
Reduce the expression (4x^(-1))/(2) by removing a factor of 2 from the numerator and denominator.
(9x^(3))/(2)-(37x)/(2)+2x^(-1)=0
Remove the negative exponent in the numerator by rewriting 2x^(-1) as (2)/(x). A negative exponent follows the rule: a^(-n)=(1)/(a^(n)).
(9x^(3))/(2)-(37x)/(2)+(2)/(x)=0
Find the LCD (least common denominator) of (9x^(3))/(2)-(37x)/(2)+(2)/(x)+0.
Least common denominator: 2x
Multiply each term in the equation by 2x in order to remove all the denominators from the equation.
(9x^(3))/(2)*2x-(37x)/(2)*2x+(2)/(x)*2x=0*2x
Simplify the left-hand side of the equation by canceling the common factors.
9x^(4)-37x^(2)+4=0*2x
Multiply 0 by 2x to get 0.
9x^(4)-37x^(2)+4=0
Substitute u=x^(2) into the equation. This will make the quadratic formula easy to use.
9u^(2)-37u+4=0_u=x^(2)
To set the left-hand side of the equation equal to 0, move all the expressions to the left-hand side.
9u^(2)-37u+4=0
In this problem -(1)/(9)*-4=4 and -(1)/(9)-4=-37, so insert -(1)/(9) as the right hand term of one factor and -4 as the right-hand term of the other factor.
(u-(1)/(9))(u-4)=0
Remove the fraction by multiplying the first term of the factor by the denominator of the second term.
(9u-1)(u-4)=0
Set each of the factors of the left-hand side of the equation equal to 0.
9u-1=0_u-4=0
Since -1 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 1 to both sides.
9u=1_u-4=0
Divide each term in the equation by 9.
(9u)/(9)=(1)/(9)_u-4=0
Simplify the left-hand side of the equation by canceling the common factors.
u=(1)/(9)_u-4=0
Set each of the factors of the left-hand side of the equation equal to 0.
u=(1)/(9)_u-4=0
Since -4 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 4 to both sides.
u=(1)/(9)_u=4
The complete solution is the set of the individual solutions.
u=(1)/(9),4
Substitute the real value of u=x^(2) back into the solved equation.
x^(2)=(1)/(9)_x^(2)=4
Solve the first equation for x.
x^(2)=(1)/(9)
Take the square root of both sides of the equation to eliminate the exponent on the left-hand side.
x=\~((1)/(9))
Split the fraction inside the radical into a separate radical expression in the numerator and the denominator. A fraction of roots is equivalent to a root of the fraction.
x=\(1)/(~(9))
Pull all perfect square roots out from under the radical. In this case, remove the 3 because it is a perfect square.
x=\(1)/(3)
First, substitute in the + portion of the \ to find the first solution.
x=(1)/(3)
Next, substitute in the - portion of the \ to find the second solution.
x=-(1)/(3)
The complete solution is the result of both the + and - portions of the solution.
x=(1)/(3),-(1)/(3)
Solve the second equation for x.
x^(2)=4
Solve the equation for x.
x=2,-2
The solution to 9x^(4)-37x^(2)+4=0 is x=(1)/(3),-(1)/(3),2,-2.
x=(1)/(3),-(1)/(3),2,-2
Answer by MathTherapy(10552)  (Show Source):
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