Find all the zeroes of the equation. −2x4−64x2+288=0
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To find the zeroes of the equation −2x^4−64x^2+288=0, we can use the quadratic formula by letting y = x^2:
-2y^2 - 64y + 288 = 0
Now we can solve for y using the quadratic formula:
y = (-(-64) ± √((-64)^2 - 4(-2)(288)))/(2(-2))
y = (64 ± √(4096 + 2304))/-4
y = (64 ± √(6400))/-4
y = (64 ± 80)/-4
For the positive root:
y = (64 + 80)/-4
y = 16
For the negative root:
y = (64 - 80)/-4
y = -4
Since y = x^2, we can now solve for x by taking the square root of both sides:
For the positive root:
x^2 = 16
x = ± √(16)
x = ± 4
For the negative root:
x^2 = -4 (no real solutions since the square of any real number cannot be negative)
Thus, the zeroes of the equation −2x^4−64x^2+288=0 are x = ± 4.