What are the solutions to the equation (x-3)^4+(x-5)^4=20 ?

The solutions to the equation (x-3)^4+(x-5)^4=20 are x\approx 2.88526…,x\approx 5.11473…

Find the zeros of (x-3)^4+(x-5)^4=20

The zeros of (x-3)^4+(x-5)^4=20 are x\approx 2.88526…,x\approx 5.11473…

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(x - 3)^{4} + (x - 5)^{4} = 20

x \approx 2.88526 \dots, x \approx 5.11473 \dots

Solution steps

(x - 3)^{4} + (x - 5)^{4} = 20

Expand

(x - 3)^{4} + (x - 5)^{4} : 2 x^{4} - 32 x^{3} + 204 x^{2} - 608 x + 706

2 x^{4} - 32 x^{3} + 204 x^{2} - 608 x + 706 = 20

Move

20

to the left side

2 x^{4} - 32 x^{3} + 204 x^{2} - 608 x + 686 = 0

Find one solution for

2 x^{4} - 32 x^{3} + 204 x^{2} - 608 x + 686 = 0

using Newton-Raphson:

x \approx 2.88526 \dots

Apply long division:

\frac{2 x ^{4} - 32 x ^{3} + 204 x ^{2} - 608 x + 686}{x - 2.88526 \dots} = 2 x^{3} - 26.22947 \dots x^{2} + 128.32108 \dots x - 237.76003 \dots

2 x^{3} - 26.22947 \dots x^{2} + 128.32108 \dots x - 237.76003 \dots \approx 0

Find one solution for

2 x^{3} - 26.22947 \dots x^{2} + 128.32108 \dots x - 237.76003 \dots = 0

using Newton-Raphson:

x \approx 5.11473 \dots

Apply long division:

\frac{2 x ^{3} - 26.22947 \dots x ^{2} + 128.32108 \dots x - 237.76003 \dots}{x - 5.11473 \dots} = 2 x^{2} - 16 x + 46.48528 \dots

2 x^{2} - 16 x + 46.48528 \dots \approx 0

Find one solution for

2 x^{2} - 16 x + 46.48528 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx 2.88526 \dots, x \approx 5.11473 \dots

5 y^{3} + 6 y^{2} + 25 y + 30 = 0

x^{3} - 6 x^{2} + 11 x - 6

y, x = y^{4}

10 x^{9} = 7 x

5 y^{4} - y^{2} + 2 y - 4 = 0

What are the solutions to the equation (x-3)^4+(x-5)^4=20 ?
The solutions to the equation (x-3)^4+(x-5)^4=20 are x\approx 2.88526…,x\approx 5.11473…
Find the zeros of (x-3)^4+(x-5)^4=20
The zeros of (x-3)^4+(x-5)^4=20 are x\approx 2.88526…,x\approx 5.11473…