Question 345388: Solve
6x^4-13x^2+5=0
Answer by haileytucki(390)   (Show Source):
You can put this solution on YOUR website! 6x^(4)-13x^(2)+5=0 (In this problem, the \ stands for +- and the ~ stands for square root)
Substitute u=x^(2) into the equation. This will make the quadratic formula easy to use.
6u^(2)-13u+5=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.
6u^(2)-13u+5=0
In this problem -(1)/(2)*-(5)/(3)=5 and -(1)/(2)-(5)/(3)=-13, so insert -(1)/(2) as the right hand term of one factor and -(5)/(3) as the right-hand term of the other factor.
(u-(1)/(2))(u-(5)/(3))=0
Remove the fraction by multiplying the first term of the factor by the denominator of the second term.
(2u-1)(3u-5)=0
Set each of the factors of the left-hand side of the equation equal to 0.
2u-1=0_3u-5=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.
2u=1_3u-5=0
Divide each term in the equation by 2.
(2u)/(2)=(1)/(2)_3u-5=0
Simplify the left-hand side of the equation by canceling the common factors.
u=(1)/(2)_3u-5=0
Set each of the factors of the left-hand side of the equation equal to 0.
u=(1)/(2)_3u-5=0
Since -5 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 5 to both sides.
u=(1)/(2)_3u=5
Divide each term in the equation by 3.
u=(1)/(2)_(3u)/(3)=(5)/(3)
Simplify the left-hand side of the equation by canceling the common factors.
u=(1)/(2)_u=(5)/(3)
The complete solution is the set of the individual solutions.
u=(1)/(2),(5)/(3)
Substitute the real value of u=x^(2) back into the solved equation.
x^(2)=(1)/(2)_x^(2)=(5)/(3)
Solve the first equation for x.
x^(2)=(1)/(2)
Take the square root of both sides of the equation to eliminate the exponent on the left-hand side.
x=\~((1)/(2))
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)/(~(2))
To rationalize the denominator of a fraction, rewrite the fraction so that the new fraction has the same value as the original and has a rational denominator. The factor to multiply by should be an expression that will eliminate the radical in the denominator. In this case, the expression that will eliminate the radical in the denominator is (~(2))/(~(2)).
x=\(1)/(~(2))*(~(2))/(~(2))
Remove the parentheses around the expression 2.
x=\(~(2))/(2)
First, substitute in the + portion of the \ to find the first solution.
x=(~(2))/(2)
Next, substitute in the - portion of the \ to find the second solution.
x=-(~(2))/(2)
The complete solution is the result of both the + and - portions of the solution.
x=(~(2))/(2),-(~(2))/(2)
Solve the second equation for x.
x^(2)=(5)/(3)
Solve the equation for x.
x=(~(15))/(3),-(~(15))/(3)
The solution to 6x^(4)-13x^(2)+5=0 is x=(~(2))/(2),-(~(2))/(2),(~(15))/(3),-(~(15))/(3).
x=(~(2))/(2),-(~(2))/(2),(~(15))/(3),-(~(15))/(3)_xAPPR0.71,-0.71,1.29,-1.29
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