O lvl A Maths: Quadratic Equations

Show that the quadratic equation k(x² - 2x) + 2 = 3x² - k has no real roots if k < -6.

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Answer:

k(x² - 2x) + 2 = 3x² - k
kx² - 2kx + 2 = 3x² - k
kx² - 2kx + 2 - 3x² + k = 0
(k - 3)x² - 2kx + (k + 2) = 0

For no real roots, b² - 4ac < 0

Hence,
(-2k)² - 4(k - 3)(k + 2) < 0
4k² - 4(k² - k - 6) < 0
k² - (k² - k - 6) < 0
k² - k² + k + 6 < 0
k < -6