Question

Determine between which consecutive intergers the real zeros of f(x)= x^4-8x^2+10 are located.

Answer 1

C

Explanation:

We are given the function:

`\displaystyle f(x) = x^4 - 8x^2 + 10`

And we want to determine between which consecutive integers are the real zeros of f located.

Because the equation is in quadratic form, we can use the quadratic formula. First, we can let u = x². Then:

`\displaystyle f(u) = u^2 - 8u + 10 = 0`

Solve using the quadratic formula:

`\displaystyle \begin{aligned} u &= (-b\pm√(b^2-4ac))/(2a) \\ \\ & = (-(-8)\pm√((-8)^2 -4(1)(10)))/(2(1)) \\ \\ & = (8\pm√(24))/(2) \\ \\ &=(8\pm2√(6))/(2)} \\ \\ & = 4\pm√(6) \end{aligned}`

Hence:

`\displaystyle u = 4+√(6) \text{ or } u = 4-√(6)`

Back-substitute:

`x^2 = 4+√(6) \text{ or } x^2 = 4-√(6)`

Solve for x:

`\displaystyle x = \pm\sqrt{4+√(6)} \text{ or } x=\pm\sqrt{4-√(6)}`

Hence, the four real solutions of x are:

`\displaytstyle x = -\sqrt{4-√(6)}, -\sqrt{4+√(6)}, \sqrt{4-√(6)}, \sqrt{4+√(6)}`

Or, approximately:

`\displaystyle x \approx -1.245, -2.540, 1.245, 2.540`

Respectively.

Therefore, our answers are both A and B, or simply C.