What are the solutions to the equation 5^2x-3-2(x^5)^2=5^2x^3 ?

The solutions to the equation 5^2x-3-2(x^5)^2=5^2x^3 are x\approx 0.12180…,x\approx 0.91286…

Find the zeros of 5^2x-3-2(x^5)^2=5^2x^3

The zeros of 5^2x-3-2(x^5)^2=5^2x^3 are x\approx 0.12180…,x\approx 0.91286…

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5^{2} x - 3 - 2 (x^{5})^{2} = 5^{2} x^{3}

x \approx 0.12180 \dots, x \approx 0.91286 \dots

Solution steps

5^{2} x - 3 - 2 (x^{5})^{2} = 5^{2} x^{3}

Expand

5^{2} x - 3 - 2 (x^{5})^{2} : 25 x - 3 - 2 x^{10}

Expand

5^{2} x^{3} : 25 x^{3}

25 x - 3 - 2 x^{10} = 25 x^{3}

Move

25 x^{3}

to the left side

25 x - 3 - 2 x^{10} - 25 x^{3} = 0

Write in the standard form

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

- 2 x^{10} - 25 x^{3} + 25 x - 3 = 0

Find one solution for

- 2 x^{10} - 25 x^{3} + 25 x - 3 = 0

using Newton-Raphson:

x \approx 0.12180 \dots

Apply long division:

\frac{- 2 x ^{10} - 25 x ^{3} + 25 x - 3}{x - 0.12180 \dots} = - 2 x^{9} - 0.24361 \dots x^{8} - 0.02967 \dots x^{7} - 0.00361 \dots x^{6} - 0.00044 \dots x^{5} - 0.00005 \dots x^{4} - 6.53234 E - 6 x^{3} - 25.00000 \dots x^{2} - 3.04518 \dots x + 24.62907 \dots

- 2 x^{9} - 0.24361 \dots x^{8} - 0.02967 \dots x^{7} - 0.00361 \dots x^{6} - 0.00044 \dots x^{5} - 0.00005 \dots x^{4} - 6.53234 E - 6 x^{3} - 25.00000 \dots x^{2} - 3.04518 \dots x + 24.62907 \dots \approx 0

Find one solution for

- 2 x^{9} - 0.24361 \dots x^{8} - 0.02967 \dots x^{7} - 0.00361 \dots x^{6} - 0.00044 \dots x^{5} - 0.00005 \dots x^{4} - 6.53234 E - 6 x^{3} - 25.00000 \dots x^{2} - 3.04518 \dots x + 24.62907 \dots = 0

using Newton-Raphson:

x \approx 0.91286 \dots

Apply long division:

\frac{- 2 x ^{9} - 0.24361 \dots x ^{8} - 0.02967 \dots x ^{7} - 0.00361 \dots x ^{6} - 0.00044 \dots x ^{5} - 0.00005 \dots x ^{4} - 6.53234 E - 6 x ^{3} - 25.00000 \dots x ^{2} - 3.04518 \dots x + 24.62907 \dots}{x - 0.91286 \dots} = - 2 x^{8} - 2.06935 \dots x^{7} - 1.91871 \dots x^{6} - 1.75515 \dots x^{5} - 1.60266 \dots x^{4} - 1.46307 \dots x^{3} - 1.33559 \dots x^{2} - 26.21922 \dots x - 26.97988 \dots

- 2 x^{8} - 2.06935 \dots x^{7} - 1.91871 \dots x^{6} - 1.75515 \dots x^{5} - 1.60266 \dots x^{4} - 1.46307 \dots x^{3} - 1.33559 \dots x^{2} - 26.21922 \dots x - 26.97988 \dots \approx 0

Find one solution for

- 2 x^{8} - 2.06935 \dots x^{7} - 1.91871 \dots x^{6} - 1.75515 \dots x^{5} - 1.60266 \dots x^{4} - 1.46307 \dots x^{3} - 1.33559 \dots x^{2} - 26.21922 \dots x - 26.97988 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx 0.12180 \dots, x \approx 0.91286 \dots

y^{8} = 1

a^{3} = 125

4 x + 3 = x^{3} + 3 x^{2} - 9 x + 12

4 x^{4} + 2 x^{2} + 1 = 0

(x^{2} - 5 x + 6) (x^{2} + 5 x + 6) - 2 x^{2} = 0

What are the solutions to the equation 5^2x-3-2(x^5)^2=5^2x^3 ?
The solutions to the equation 5^2x-3-2(x^5)^2=5^2x^3 are x\approx 0.12180…,x\approx 0.91286…
Find the zeros of 5^2x-3-2(x^5)^2=5^2x^3
The zeros of 5^2x-3-2(x^5)^2=5^2x^3 are x\approx 0.12180…,x\approx 0.91286…