What are the solutions to the equation 2(1+x)^{10}=(1+x)^{16} ?

The solutions to the equation 2(1+x)^{10}=(1+x)^{16} are x=-1,x\approx 0.12246…,x\approx-2.12246…

Find the zeros of 2(1+x)^{10}=(1+x)^{16}

The zeros of 2(1+x)^{10}=(1+x)^{16} are x=-1,x\approx 0.12246…,x\approx-2.12246…

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2 (1 + x)^{10} = (1 + x)^{16}

x = - 1, x \approx 0.12246 \dots, x \approx - 2.12246 \dots

Solution steps

2 (1 + x)^{10} = (1 + x)^{16}

Expand

2 (1 + x)^{10} : 2 + 20 x + 90 x^{2} + 240 x^{3} + 420 x^{4} + 504 x^{5} + 420 x^{6} + 240 x^{7} + 90 x^{8} + 20 x^{9} + 2 x^{10}

Expand

(1 + x)^{16} : 1 + 16 x + 120 x^{2} + 560 x^{3} + 1820 x^{4} + 4368 x^{5} + 8008 x^{6} + 11440 x^{7} + 12870 x^{8} + 11440 x^{9} + 8008 x^{10} + 4368 x^{11} + 1820 x^{12} + 560 x^{13} + 120 x^{14} + 16 x^{15} + x^{16}

2 + 20 x + 90 x^{2} + 240 x^{3} + 420 x^{4} + 504 x^{5} + 420 x^{6} + 240 x^{7} + 90 x^{8} + 20 x^{9} + 2 x^{10} = 1 + 16 x + 120 x^{2} + 560 x^{3} + 1820 x^{4} + 4368 x^{5} + 8008 x^{6} + 11440 x^{7} + 12870 x^{8} + 11440 x^{9} + 8008 x^{10} + 4368 x^{11} + 1820 x^{12} + 560 x^{13} + 120 x^{14} + 16 x^{15} + x^{16}

Switch sides

1 + 16 x + 120 x^{2} + 560 x^{3} + 1820 x^{4} + 4368 x^{5} + 8008 x^{6} + 11440 x^{7} + 12870 x^{8} + 11440 x^{9} + 8008 x^{10} + 4368 x^{11} + 1820 x^{12} + 560 x^{13} + 120 x^{14} + 16 x^{15} + x^{16} = 2 + 20 x + 90 x^{2} + 240 x^{3} + 420 x^{4} + 504 x^{5} + 420 x^{6} + 240 x^{7} + 90 x^{8} + 20 x^{9} + 2 x^{10}

Subtract

2 + 20 x + 90 x^{2} + 240 x^{3} + 420 x^{4} + 504 x^{5} + 420 x^{6} + 240 x^{7} + 90 x^{8} + 20 x^{9} + 2 x^{10}

from both sides

1 + 16 x + 120 x^{2} + 560 x^{3} + 1820 x^{4} + 4368 x^{5} + 8008 x^{6} + 11440 x^{7} + 12870 x^{8} + 11440 x^{9} + 8008 x^{10} + 4368 x^{11} + 1820 x^{12} + 560 x^{13} + 120 x^{14} + 16 x^{15} + x^{16} - (2 + 20 x + 90 x^{2} + 240 x^{3} + 420 x^{4} + 504 x^{5} + 420 x^{6} + 240 x^{7} + 90 x^{8} + 20 x^{9} + 2 x^{10}) = 2 + 20 x + 90 x^{2} + 240 x^{3} + 420 x^{4} + 504 x^{5} + 420 x^{6} + 240 x^{7} + 90 x^{8} + 20 x^{9} + 2 x^{10} - (2 + 20 x + 90 x^{2} + 240 x^{3} + 420 x^{4} + 504 x^{5} + 420 x^{6} + 240 x^{7} + 90 x^{8} + 20 x^{9} + 2 x^{10})

Simplify

x^{16} + 16 x^{15} + 120 x^{14} + 560 x^{13} + 1820 x^{12} + 4368 x^{11} + 8006 x^{10} + 11420 x^{9} + 12780 x^{8} + 11200 x^{7} + 7588 x^{6} + 3864 x^{5} + 1400 x^{4} + 320 x^{3} + 30 x^{2} - 4 x - 1 = 0

Factor

x^{16} + 16 x^{15} + 120 x^{14} + 560 x^{13} + 1820 x^{12} + 4368 x^{11} + 8006 x^{10} + 11420 x^{9} + 12780 x^{8} + 11200 x^{7} + 7588 x^{6} + 3864 x^{5} + 1400 x^{4} + 320 x^{3} + 30 x^{2} - 4 x - 1 : (x + 1)^{10} (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x - 1)

(x + 1)^{10} (x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x - 1) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

x + 1 = 0 or x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x - 1 = 0

Solve

x + 1 = 0 : x = - 1

Solve

x^{6} + 6 x^{5} + 15 x^{4} + 20 x^{3} + 15 x^{2} + 6 x - 1 = 0 : x \approx 0.12246 \dots, x \approx - 2.12246 \dots

The solutions are

x = - 1, x \approx 0.12246 \dots, x \approx - 2.12246 \dots

a^{4} = 5 (4) - 1

y^{3} = 1000

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

- x^{3} + 15 x^{2} + 91 x + 51 = 0

60 x^{4} - 120 x^{3} = 0

What are the solutions to the equation 2(1+x)^{10}=(1+x)^{16} ?
The solutions to the equation 2(1+x)^{10}=(1+x)^{16} are x=-1,x\approx 0.12246…,x\approx-2.12246…
Find the zeros of 2(1+x)^{10}=(1+x)^{16}
The zeros of 2(1+x)^{10}=(1+x)^{16} are x=-1,x\approx 0.12246…,x\approx-2.12246…