What are the solutions to the equation 3x+5x^6=4.5(x+6) ?

The solutions to the equation 3x+5x^6=4.5(x+6) are x\approx-1.30799…,x\approx 1.34049…

Find the zeros of 3x+5x^6=4.5(x+6)

The zeros of 3x+5x^6=4.5(x+6) are x\approx-1.30799…,x\approx 1.34049…

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3 x + 5 x^{6} = 4.5 (x + 6)

x \approx - 1.30799 \dots, x \approx 1.34049 \dots

Solution steps

3 x + 5 x^{6} = 4.5 (x + 6)

Multiply both sides by

10

30 x + 50 x^{6} = 45 (x + 6)

Expand

45 (x + 6) : 45 x + 270

30 x + 50 x^{6} = 45 x + 270

Move

270

to the left side

30 x + 50 x^{6} - 270 = 45 x

Move

45 x

to the left side

50 x^{6} - 15 x - 270 = 0

Find one solution for

50 x^{6} - 15 x - 270 = 0

using Newton-Raphson:

x \approx - 1.30799 \dots

Apply long division:

\frac{50 x ^{6} - 15 x - 270}{x + 1.30799 \dots} = 50 x^{5} - 65.39958 \dots x^{4} + 85.54210 \dots x^{3} - 111.88836 \dots x^{2} + 146.34904 \dots x - 206.42333 \dots

50 x^{5} - 65.39958 \dots x^{4} + 85.54210 \dots x^{3} - 111.88836 \dots x^{2} + 146.34904 \dots x - 206.42333 \dots \approx 0

Find one solution for

50 x^{5} - 65.39958 \dots x^{4} + 85.54210 \dots x^{3} - 111.88836 \dots x^{2} + 146.34904 \dots x - 206.42333 \dots = 0

using Newton-Raphson:

x \approx 1.34049 \dots

Apply long division:

\frac{50 x ^{5} - 65.39958 \dots x ^{4} + 85.54210 \dots x ^{3} - 111.88836 \dots x ^{2} + 146.34904 \dots x - 206.42333 \dots}{x - 1.34049 \dots} = 50 x^{4} + 1.62510 \dots x^{3} + 87.72055 \dots x^{2} + 5.70049 \dots x + 153.99052 \dots

50 x^{4} + 1.62510 \dots x^{3} + 87.72055 \dots x^{2} + 5.70049 \dots x + 153.99052 \dots \approx 0

Find one solution for

50 x^{4} + 1.62510 \dots x^{3} + 87.72055 \dots x^{2} + 5.70049 \dots x + 153.99052 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx - 1.30799 \dots, x \approx 1.34049 \dots

0 = 4 x^{3} - 9 x^{2} + 6 x - 1

x^{4} - 3 x^{2} + 2 x - 5 = 0

0 = x^{3} - 2 x + 1

- x^{4} + x^{3} + 45 x^{2} - 25 x - 500 = 0

x = \frac{2}{3} π r^{3}, r

What are the solutions to the equation 3x+5x^6=4.5(x+6) ?
The solutions to the equation 3x+5x^6=4.5(x+6) are x\approx-1.30799…,x\approx 1.34049…
Find the zeros of 3x+5x^6=4.5(x+6)
The zeros of 3x+5x^6=4.5(x+6) are x\approx-1.30799…,x\approx 1.34049…