Question

Find all real and/or imaginary solutions. x4+5x2−14=0 (1 point) Responses x=±2–√i or x=±7–√ x=±2–√i or x=±7–√ x=±7–√i or x=±2–√ x=±7–√i or x=±2–√ x=±6–√i or x=±3–√ x=±6–√i or x=±3–√ x=±3–√i or x=±6–√ x=±3–√i or x=±6–√

Answer 1

Let us solve the equation step by step:

Given: x^4 + 5x^2 − 14 = 0.

This is a quadratic form in terms of x^2. Let y = x^2.

Then the equation becomes: y^2 + 5y − 14 = 0.

Now we solve for y by factoring or using the quadratic formula. Looking to factor, we find factors of -14 that add up to 5:

(y + 7)(y - 2) = 0.

So either y + 7 = 0 or y - 2 = 0.

If y + 7 = 0, then y = -7.
If y - 2 = 0, then y = 2.

Now we need to find the values for x, recalling that y = x^2.

For y = -7, since we can't have a real number squared being negative, we require imaginary solutions:

x^2 = -7.
x = ±sqrt(-7).
x = ±i*sqrt(7).

For y = 2:

x^2 = 2.
x = ±sqrt(2).

Thus, the solutions for the equation x^4 + 5x^2 − 14 = 0 are:

x = ±i*sqrt(7) or x = ±sqrt(2).

None of the responses you provided precisely match the correct solutions, but the correct options would be:

x = ±sqrt(7)i or x = ±sqrt(2).