What are the solutions to the equation 12x^4-32x^3+13x^2+8x+1=0 ?

The solutions to the equation 12x^4-32x^3+13x^2+8x+1=0 are x\approx 1.13204…,x\approx 1.89932…

Find the zeros of 12x^4-32x^3+13x^2+8x+1=0

The zeros of 12x^4-32x^3+13x^2+8x+1=0 are x\approx 1.13204…,x\approx 1.89932…

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12 x^{4} - 32 x^{3} + 13 x^{2} + 8 x + 1 = 0

x \approx 1.13204 \dots, x \approx 1.89932 \dots

Solution steps

12 x^{4} - 32 x^{3} + 13 x^{2} + 8 x + 1 = 0

Find one solution for

12 x^{4} - 32 x^{3} + 13 x^{2} + 8 x + 1 = 0

using Newton-Raphson:

x \approx 1.13204 \dots

Apply long division:

\frac{12 x ^{4} - 32 x ^{3} + 13 x ^{2} + 8 x + 1}{x - 1.13204 \dots} = 12 x^{3} - 18.41541 \dots x^{2} - 7.84714 \dots x - 0.88335 \dots

12 x^{3} - 18.41541 \dots x^{2} - 7.84714 \dots x - 0.88335 \dots \approx 0

Find one solution for

12 x^{3} - 18.41541 \dots x^{2} - 7.84714 \dots x - 0.88335 \dots = 0

using Newton-Raphson:

x \approx 1.89932 \dots

Apply long division:

\frac{12 x ^{3} - 18.41541 \dots x ^{2} - 7.84714 \dots x - 0.88335 \dots}{x - 1.89932 \dots} = 12 x^{2} + 4.37642 \dots x + 0.46508 \dots

12 x^{2} + 4.37642 \dots x + 0.46508 \dots \approx 0

Find one solution for

12 x^{2} + 4.37642 \dots x + 0.46508 \dots = 0

using Newton-Raphson:

No Solution for

x \in R

The solutions are

x \approx 1.13204 \dots, x \approx 1.89932 \dots

a^{3} = 3^{9} \times 1 2^{3} \times 2 1^{6}

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

y, 2 x^{3} y^{1} = y (2 x^{2} - y^{2})

x^{10} + 6 x^{5} - 16 = 0

x^{3} - 7 x - 6

What are the solutions to the equation 12x^4-32x^3+13x^2+8x+1=0 ?
The solutions to the equation 12x^4-32x^3+13x^2+8x+1=0 are x\approx 1.13204…,x\approx 1.89932…
Find the zeros of 12x^4-32x^3+13x^2+8x+1=0
The zeros of 12x^4-32x^3+13x^2+8x+1=0 are x\approx 1.13204…,x\approx 1.89932…