What are the solutions to the equation 5x+21=10x^{19} ?

The solutions to the equation 5x+21=10x^{19} are x\approx 1.05212…

Find the zeros of 5x+21=10x^{19}

The zeros of 5x+21=10x^{19} are x\approx 1.05212…

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5 x + 21 = 10 x^{19}

x \approx 1.05212 \dots

Solution steps

5 x + 21 = 10 x^{19}

Switch sides

10 x^{19} = 5 x + 21

Move

21

to the left side

10 x^{19} - 21 = 5 x

Move

5 x

to the left side

10 x^{19} - 21 - 5 x = 0

Write in the standard form

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

10 x^{19} - 5 x - 21 = 0

Find one solution for

10 x^{19} - 5 x - 21 = 0

using Newton-Raphson:

x \approx 1.05212 \dots

Apply long division:

\frac{10 x ^{19} - 5 x - 21}{x - 1.05212 \dots} = 10 x^{18} + 10.52128 \dots x^{17} + 11.06973 \dots x^{16} + 11.64678 \dots x^{15} + 12.25391 \dots x^{14} + 12.89268 \dots x^{13} + 13.56476 \dots x^{12} + 14.27187 \dots x^{11} + 15.01583 \dots x^{10} + 15.79858 \dots x^{9} + 16.62214 \dots x^{8} + 17.48862 \dots x^{7} + 18.40027 \dots x^{6} + 19.35945 \dots x^{5} + 20.36863 \dots x^{4} + 21.43041 \dots x^{3} + 22.54754 \dots x^{2} + 23.72290 \dots x + 19.95954 \dots

10 x^{18} + 10.52128 \dots x^{17} + 11.06973 \dots x^{16} + 11.64678 \dots x^{15} + 12.25391 \dots x^{14} + 12.89268 \dots x^{13} + 13.56476 \dots x^{12} + 14.27187 \dots x^{11} + 15.01583 \dots x^{10} + 15.79858 \dots x^{9} + 16.62214 \dots x^{8} + 17.48862 \dots x^{7} + 18.40027 \dots x^{6} + 19.35945 \dots x^{5} + 20.36863 \dots x^{4} + 21.43041 \dots x^{3} + 22.54754 \dots x^{2} + 23.72290 \dots x + 19.95954 \dots \approx 0

Find one solution for

10 x^{18} + 10.52128 \dots x^{17} + 11.06973 \dots x^{16} + 11.64678 \dots x^{15} + 12.25391 \dots x^{14} + 12.89268 \dots x^{13} + 13.56476 \dots x^{12} + 14.27187 \dots x^{11} + 15.01583 \dots x^{10} + 15.79858 \dots x^{9} + 16.62214 \dots x^{8} + 17.48862 \dots x^{7} + 18.40027 \dots x^{6} + 19.35945 \dots x^{5} + 20.36863 \dots x^{4} + 21.43041 \dots x^{3} + 22.54754 \dots x^{2} + 23.72290 \dots x + 19.95954 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solution is

x \approx 1.05212 \dots

x^{4} - 2 x^{3} - 8 x^{2} + 18 x - 9 = 0

7 x^{3} - 35 x^{2} - 252 x = 0

y^{3} = 64 y

2 = y^{3} + 2 y - 1

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

What are the solutions to the equation 5x+21=10x^{19} ?
The solutions to the equation 5x+21=10x^{19} are x\approx 1.05212…
Find the zeros of 5x+21=10x^{19}
The zeros of 5x+21=10x^{19} are x\approx 1.05212…