Question

Find all the zeroes of the function. y = x^4 - 6x^2 + 8

Answer 1

$y=x^4-6x^2+8\\\\y=x^4-6x^2+9-1\\\\y=(x^2)^2-2\cdot3\cdot x^2+3^2-1\qquad\qquad \big[\text{use }a^2-2ab+b^2=(a-b)^2\big]\\\\y=(x^2-3)^2-1\\\\ y=(x^2-3)^2-1^2\qquad\qquad\qquad\quad\qquad\big[a^2-b^2=(a-b)(a+b)\big]\\\\y=(x^2-3-1)(x^2-3+1)\\\\y=(x^2-4)(x^2-2)\\\\y=(x^2-2^2)(x^2-\sqrt{2}^2)\\\\y=(x+2)(x-2)(x+\sqrt{2})(x-\sqrt{2})\\\\\\\boxed{x=-2\qquad\vee\qquad x=2\qquad\vee\qquad x=-\sqrt{2}\qquad\vee\qquad x=\sqrt{2}}$