Question

k + 6 is the mean proportion between k − 2 and k². Find the value of k.

Answer 1

k + 6 is the mean proportion between k − 2 and k².

⇒ (k + 6)² = (k − 2) (k²)

⇒ (k + 6)² = k²(k − 2)

Left side:

(k + 6)² = k² + 12k + 36

Right side:

k²(k − 2) = k³ − 2k²

So we get,

⇒ k² + 12k + 36 = k³ − 2k²

⇒ k³ − 2k² − k² − 12k − 36 = 0

⇒ k³ − 3k² − 12k − 36 = 0

⇒ (k³ − 3k²) − (12k + 36) = 0

⇒ k²(k − 3) − 12(k + 3) = 0

This grouping does not factor cleanly, so we try a trial value

Check if k = 6 is a root:

k³ − 3k² − 12k − 36

= 6³ − 3(6²) − 12(6) − 36

= 216 − 108 − 72 − 36

= 0

∴ The value of k is 6.