Question

Solve the following by reducing them to quadratic equations:

(i) x⁴ – 26x² + 25 = 0

(ii) z⁴ – 10z² + 9 = 0

Answer 1

(i) Given x⁴ – 26x² + 25 = 0

Putting, x² = y, the given equation reduces to the form y² – 26y + 25  = 0

⇒ y² – 25y – y + 25 = 0

⇒ y(y – 25) – 1(y – 25) = 0

⇒ (y – 25)(y – 1) = 0

⇒ y – 25 = 0  or  y – 1 = 0

⇒ y = 25  or  y = 1

∴   x² = 25

⇒  x  = ± 5

or    x² = 1

x = ± 1

Hence, the required roots are ± 5, ± 1.

(ii) Given equation z⁴ – 10z² + 9 = 0

Putting  z² = x, then given equation reduces to the form x² – 10x + 9 = 0

⇒   x² – 9x – x + 9 = 0

⇒  x(x – 9) – 1(x – 9) = 0

⇒  (x – 9)(x – 1) = 0

⇒ x – 9 = 0 or  x – 1 = 1

⇒  x = 9  or  x = 1

But z² = x

∴    z² = 9

⇒ z = ± 3

Or   z² = 1

z = ± 1

Hence, the required roots are ± 3, ± 1.