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Popular >

x^4-2x^3+2=-x+7/4

Solution

x^{4} - 2 x^{3} + 2 = - x + \frac{7}{4}

Solution

x \approx - 0.36602 \dots, x \approx - 0.36602 \dots, x \approx 1.36602 \dots, x = \frac{1351}{989}

+1

Decimal Notation

x = - 0.36602 \dots, x = - 0.36602 \dots, x = 1.36602 \dots, x = 1.36602 \dots

Solution steps

x^{4} - 2 x^{3} + 2 = - x + \frac{7}{4}

Multiply both sides by

4

4 x^{4} - 8 x^{3} + 8 = - 4 x + 7

Move

7

to the left side

4 x^{4} - 8 x^{3} + 1 = - 4 x

Move

4 x

to the left side

4 x^{4} - 8 x^{3} + 1 + 4 x = 0

Write in the standard form

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

4 x^{4} - 8 x^{3} + 4 x + 1 = 0

Find one solution for

4 x^{4} - 8 x^{3} + 4 x + 1 = 0

using Newton-Raphson:

x \approx - 0.36602 \dots

Apply long division:

\frac{4 x ^{4} - 8 x ^{3} + 4 x + 1}{x + 0.36602 \dots} = 4 x^{3} - 9.46409 \dots x^{2} + 3.46409 \dots x + 2.73205 \dots

4 x^{3} - 9.46409 \dots x^{2} + 3.46409 \dots x + 2.73205 \dots \approx 0

Find one solution for

4 x^{3} - 9.46409 \dots x^{2} + 3.46409 \dots x + 2.73205 \dots = 0

using Newton-Raphson:

x \approx - 0.36602 \dots

Apply long division:

\frac{4 x ^{3} - 9.46409 \dots x ^{2} + 3.46409 \dots x + 2.73205 \dots}{x + 0.36602 \dots} = 4 x^{2} - 10.92820 \dots x + 7.46410 \dots

4 x^{2} - 10.92820 \dots x + 7.46410 \dots \approx 0

Find one solution for

4 x^{2} - 10.92820 \dots x + 7.46410 \dots = 0

using Newton-Raphson:

x \approx 1.36602 \dots

Apply long division:

\frac{4 x ^{2} - 10.92820 \dots x + 7.46410 \dots}{x - 1.36602 \dots} = 4 x - 5.46410 \dots

4 x - 5.46410 \dots \approx 0

x \approx 1.36602 \dots

Approximate to algebraic form

x = \frac{1351}{989}

The solutions are

x \approx - 0.36602 \dots, x \approx - 0.36602 \dots, x \approx 1.36602 \dots, x = \frac{1351}{989}

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