Question 502859: 9x^(4)-18x^(2)+5=0
Answer by persian52(161)   (Show Source):
You can put this solution on YOUR website! 9x^(4)-18x^(2)+5=0
Substitute u=x^(2) into the equation. This will make the quadratic formula easy to use.
9u^(2)-18u+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.
9u^(2)-18u+5=0
Use the quadratic formula to find the solutions. In this case, the values are a=9, b=-18, and c=5.
u=(-b\~(b^(2)-4ac))/(2a) where au^(2)+bu+c=0
Substitute in the values of a=9, b=-18, and c=5.
u=(-(-18)\~((-18)^(2)-4(9)(5)))/(2(9))
Multiply -1 by the -18 inside the parentheses.
u=(18\~((-18)^(2)-4(9)(5)))/(2(9))
Simplify the section inside the radical.
u=(18\12)/(2(9))
Simplify the denominator of the quadratic formula.
u=(18\12)/(18)
First, solve the + portion of \.
u=(18+12)/(18)
Simplify the expression to solve for the + portion of the \.
u=(5)/(3)
Next, solve the - portion of \.
u=(18-12)/(18)
Simplify the expression to solve for the - portion of the \.
u=(1)/(3)
The final answer is the combination of both solutions.
u=(5)/(3),(1)/(3)
Substitute the real value of u=x^(2) back into the solved equation.
x^(2)=(5)/(3)_x^(2)=(1)/(3)
Solve the first equation for x.
x^(2)=(5)/(3)
Take the square root of both sides of the equation to eliminate the exponent on the left-hand side.
x=\~((5)/(3))
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=\(~(5))/(~(3))
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 will eliminate the radical in the denominator is (~(3))/(~(3)).
x=\(~(5))/(~(3))*(~(3))/(~(3))
Multiply (5) by (3) to get (5)(3).
x=\(~((5)(3)))/(3)
Multiply 5 by 3 to get 15.
x=\(~((15)))/(3)
Remove the parentheses around the expression 15.
x=\(~(15))/(3)
First, substitute in the + portion of the \ to find the first solution.
x=(~(15))/(3)
Next, substitute in the - portion of the \ to find the second solution.
x=-(~(15))/(3)
The complete solution is the result of both the + and - portions of the solution.
x=(~(15))/(3),-(~(15))/(3)
Solve the second equation for x.
x^(2)=(1)/(3)
Solve the equation for x.
x=(~(3))/(3),-(~(3))/(3)
The solution to 9x^(4)-18x^(2)+5=0 is x=(~(15))/(3),-(~(15))/(3),(~(3))/(3),-(~(3))/(3).
x=(~(15))/(3),-(~(15))/(3),(~(3))/(3),-(~(3))/(3)
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