First slide

Theory of equations

Question

The number of distinct real roots of the
equation
$(x + 3)^{4} + (x + 5)^{4} = 16 (1)$
is

Moderate

A

1
B

2
C

3
D

4

Solution

Put $x + 4 = t,$ so that $(1)$ becomes
$\begin{array}{l} (t - 1)^{4} + (t + 1)^{4} = 16 \\ \Rightarrow 2 (t^{4} + 6 t^{2} + 1) = 16 \\ \Rightarrow t^{4} + 6 t^{2} - 7 = 0 \\ \Rightarrow t^{2} = 1, - 7 \Rightarrow t = \pm 1, \sqrt{7} i \end{array}$
Thus, the equation (1) has two real roots.

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