3x^2(x-4)^2<32-5(x-2)^2

Lösung

3 x^{2} (x - 4)^{2} < 32 - 5 (x - 2)^{2}

\frac{2 ( 3 - 2 3 )}{3} < x < 1 or 3 < x < \frac{2 ( 3 + 2 3 )}{3}

Intervall-Notation

(\frac{2 ( 3 - 2 3 )}{3}, 1) \cup (3, \frac{2 ( 3 + 2 3 )}{3})

Dezimale

- 0.30940 \dots < x < 1 or 3 < x < 4.30940 \dots

Schritte zur Lösung

3 x^{2} (x - 4)^{2} < 32 - 5 (x - 2)^{2}

Rewrite in standard form

3 x^{4} - 24 x^{3} + 53 x^{2} - 20 x - 12 < 0

Faktor

3 x^{4} - 24 x^{3} + 53 x^{2} - 20 x - 12 : (x - 1) (x - 3) (x - \frac{2 ( 3 + 2 3 )}{3}) (x - \frac{2 ( 3 - 2 3 )}{3})

(x - 1) (x - 3) (x - \frac{2 ( 3 + 2 3 )}{3}) (x - \frac{2 ( 3 - 2 3 )}{3}) < 0

Identifiziere die Intervalle

\frac{2 ( 3 - 2 3 )}{3} < x < 1 or 3 < x < \frac{2 ( 3 + 2 3 )}{3}

x^{2} - 18 > 0

2 x^{2} + 3 x < 4

\frac{4}{( x - 2 )} > \frac{3}{( x + 5 )}

2 x^{3} - 3 x^{2} - 8 x - 3 \leq 0

3 x + 4 < x^{2} + 2