Solve the following by reducing them to quadratic equations: z^4 … - Vidyadip

Question

Solve the following by reducing them to quadratic equations:
$z^4 - 10z^2 + 9 = 0.$

✓

Answer

Given equation $z^4 - 10z^2 + 9 = 0$
Putting $z^2 = x$, then given equation reduces to the form $x^2 - 10x + 9 = 0$
$\Rightarrow x^2 - 9x - x + 9 = 0$
$\Rightarrow x(x - 9) -1(x - 9) = 0$
$\Rightarrow (x - 9) (x - 1) = 0$
$\Rightarrow x - 9 = 0$ or $x - 1 = 1$
$\Rightarrow x = 9$ or $x = 1$
But $z^2 = x$
$\therefore z^2 = 9$
$\Rightarrow z = ±3$
or
$z^2 = 1$
$z = ±1$
Hence, the required roots are $±3, ±1.$

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