The value of x is [tex]x=1, x=-1[/tex]
Explanation:
The expression is [tex]\sqrt{2 x^{2}-1}=x[/tex]
Square both sides of the equation, we get,
[tex]2 x^{2}-1=x^{4}[/tex]
Switch sides,
[tex]x^{4}=2 x^{2}-1[/tex]
Subtracting both sides by [tex]2 x^{2}[/tex],
[tex]x^{4}-2 x^{2}=-1[/tex]
Adding both sides by 1,
[tex]x^{4}-2 x^{2}+1=0[/tex]
Let us rewrite the equation as [tex]u=x^{2}[/tex] and [tex]u^{2}=x^{4}[/tex]
Thus, we have,
[tex]u^{2}-2 u+1=0[/tex]
Simplifying using quadratic formula,
[tex]u=\frac{-(-2) \pm \sqrt{(-2)^{2}-4 \cdot 1 \cdot 1}}{2 \cdot 1}[/tex]
Simplifying, we get,
[tex]u=\frac{-(-2)}{2 \cdot 1}[/tex]
[tex]u=1[/tex]
Substituting [tex]u=1[/tex] in [tex]u=x^{2}[/tex], we get,
[tex]x^{2} =1[/tex]
[tex]x=1, x=-1[/tex]
Thus, the value of x is [tex]x=1, x=-1[/tex]
