To solve the equation ( 9x^4 - 10x^2 + 1 = 0 ), we can use a substitution method to simplify it. Let's set ( y = x^2 ). Then, we can rewrite the equation in terms of ( y ):
[
9y^2 - 10y + 1 = 0
]
Now, we will use the quadratic formula to solve for ( y ):
[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
In this case, ( a = 9 ), ( b = -10 ), and ( c = 1 ). Now, we can calculate the discriminant:
[
b^2 - 4ac = (-10)^2 - 4 \cdot 9 \cdot 1 = 100 - 36 = 64
]
Now applying the quadratic formula:
[
y = \frac{-(-10) \pm \sqrt{64}}{2 \cdot 9} = \frac{10 \pm 8}{18}
]
This gives us two possible solutions for ( y ):
- ( y = \frac{10 + 8}{18} = \frac{18}{18} = 1 )
- ( y = \frac{10 - 8}{18} = \frac{2}{18} = \frac{1}{9} )
Now we substitute back ( y = x^2 ):
For ( y = 1 ):
[
x^2 = 1 \implies x = \pm 1
]
For ( y = \frac{1}{9} ):
[
x^2 = \frac{1}{9} \implies x = \pm \frac{1}{3}
]
Thus, the complete set of solutions for the original equation ( 9x^4 - 10x^2 + 1 = 0 ) is:
[
x = 1, ; x = -1, ; x = \frac{1}{3}, ; x = -\frac{1}{3}
]
