(x^2-5x+4)^2-5(x^2-5x)-18=0

Solution

(x^{2} - 5 x + 4)^{2} - 5 (x^{2} - 5 x) - 18 = 0

x \approx - 0.10989 \dots, x \approx 0.86035 \dots, x \approx 4.13964 \dots, x \approx 5.10989 \dots

Solution steps

(x^{2} - 5 x + 4)^{2} - 5 (x^{2} - 5 x) - 18 = 0

Expand

(x^{2} - 5 x + 4)^{2} - 5 (x^{2} - 5 x) - 18 : x^{4} - 10 x^{3} + 28 x^{2} - 15 x - 2

x^{4} - 10 x^{3} + 28 x^{2} - 15 x - 2 = 0

Find one solution for

x^{4} - 10 x^{3} + 28 x^{2} - 15 x - 2 = 0

using Newton-Raphson:

x \approx - 0.10989 \dots

Apply long division:

\frac{x ^{4} - 10 x ^{3} + 28 x ^{2} - 15 x - 2}{x + 0.10989 \dots} = x^{3} - 10.10989 \dots x^{2} + 29.11102 \dots x - 18.19916 \dots

x^{3} - 10.10989 \dots x^{2} + 29.11102 \dots x - 18.19916 \dots \approx 0

Find one solution for

x^{3} - 10.10989 \dots x^{2} + 29.11102 \dots x - 18.19916 \dots = 0

using Newton-Raphson:

x \approx 0.86035 \dots

Apply long division:

\frac{x ^{3} - 10.10989 \dots x ^{2} + 29.11102 \dots x - 18.19916 \dots}{x - 0.86035 \dots} = x^{2} - 9.24954 \dots x + 21.15316 \dots

x^{2} - 9.24954 \dots x + 21.15316 \dots \approx 0

Find one solution for

x^{2} - 9.24954 \dots x + 21.15316 \dots = 0

using Newton-Raphson:

x \approx 4.13964 \dots

Apply long division:

\frac{x ^{2} - 9.24954 \dots x + 21.15316 \dots}{x - 4.13964 \dots} = x - 5.10989 \dots

x - 5.10989 \dots \approx 0

x \approx 5.10989 \dots

The solutions are

x \approx - 0.10989 \dots, x \approx 0.86035 \dots, x \approx 4.13964 \dots, x \approx 5.10989 \dots

x^{3} - 3 x^{2} - 4 x + 8 = 0

x^{3} + x - 5 = 0

16 x^{3} + 23 x^{2} + 198 x - 209 = 0

x^{3} = \frac{- 125}{216}

x^{3} + 2 x^{2} + 10 x = 20