Solve x^4-x^2+1 - ScanMath

Question

Find the roots

x \in / R

Evaluate

x^{4} - x^{2} + 1

To find the roots of the expression,set the expression equal to 0

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

Solve the equation using substitution t = x^{2}

t^{2} - t + 1 = 0

Substitute a= 1,b= - 1 and c= 1 into the quadratic formula t = \frac{- b \pm b ^{2} - 4 a c}{2 a}

t = \frac{1 \pm ( - 1 ) ^{2} - 4}{2}

Simplify the expression

More Steps

Evaluate

(- 1)^{2} - 4

Evaluate the power

1 - 4

Subtract the numbers

- 3

t = \frac{1 \pm - 3}{2}

Simplify the radical expression

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Evaluate

- 3

Evaluate the power

3 \times - 1

Evaluate the power

3 \times i

t = \frac{1 \pm 3 \times i}{2}

Separate the equation into 2 possible cases

t = \frac{1 + 3 \times i}{2} t = \frac{1 - 3 \times i}{2}

Simplify the expression

t = \frac{1}{2} + \frac{3}{2} i t = \frac{1 - 3 \times i}{2}

Simplify the expression

t = \frac{1}{2} + \frac{3}{2} i t = \frac{1}{2} - \frac{3}{2} i

Substitute back

x^{2} = \frac{1}{2} + \frac{3}{2} i x^{2} = \frac{1}{2} - \frac{3}{2} i

Solve the equation for x

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Substitute back

x^{2} = \frac{1}{2} + \frac{3}{2} i

Simplify

x = \frac{1}{2} + \frac{3}{2} i

Rewrite the complex number in polar form

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Evaluate

\frac{1}{2} + \frac{3}{2} i

Determine the modulus and the argument of the complex number

r = (\frac{1}{2})^{2} + (\frac{3}{2})^{2} θ = arctan \frac{\frac{3}{2}}{\frac{1}{2}}

Calculate

r = 1 θ = arctan \frac{\frac{3}{2}}{\frac{1}{2}}

Convert to radians

r = 1 θ = \frac{π}{3}

Substitute the given values into the formula r(cosθ+isinθ)

1 \times (cos (\frac{π}{3}) + i sin (\frac{π}{3}))

x = 1 \times (cos (\frac{π}{3}) + i sin (\frac{π}{3}))

Calculate the nth roots of a complex r (cos (θ) + i \times sin (θ),using n z = n r (cos \frac{θ + 2 kπ}{n} + i sin \frac{θ + 2 kπ}{n})

x = 1 \times (cos (\frac{\frac{π}{3} + 2 kπ}{2}) + i sin (\frac{\frac{π}{3} + 2 kπ}{2}))

Simplify

x = cos (\frac{\frac{π}{3} + 2 kπ}{2}) + i sin (\frac{\frac{π}{3} + 2 kπ}{2})

Since n = 2,substitute k = 0, 1 into the expression

x_{1} = cos (\frac{\frac{π}{3} + 2 \times 0 \times π}{2}) + i sin (\frac{\frac{π}{3} + 2 \times 0 \times π}{2}) x_{2} = cos (\frac{\frac{π}{3} + 2 \times 1 \times π}{2}) + i sin (\frac{\frac{π}{3} + 2 \times 1 \times π}{2})

Calculate

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Evaluate

\frac{\frac{π}{3} + 2 \times 0 \times π}{2}

Any expression multiplied by 0 equals 0

\frac{\frac{π}{3} + 0}{2}

Removing 0 doesn't change the value,so remove it from the expression

\frac{\frac{π}{3}}{2}

Rewrite the expression

\frac{π}{3} \times \frac{1}{2}

To multiply the fractions,multiply the numerators and denominators separately

\frac{π}{3 \times 2}

Multiply the numbers

\frac{π}{6}

x_{1} = cos (\frac{π}{6}) + i sin (\frac{π}{6}) x_{2} = cos (\frac{\frac{π}{3} + 2 \times 1 \times π}{2}) + i sin (\frac{\frac{π}{3} + 2 \times 1 \times π}{2})

Calculate

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Evaluate

\frac{\frac{π}{3} + 2 \times 1 \times π}{2}

Multiply the terms

\frac{\frac{π}{3} + 2 π}{2}

Calculate

\frac{\frac{7 π}{3}}{2}

Rewrite the expression

\frac{7 π}{3} \times \frac{1}{2}

To multiply the fractions,multiply the numerators and denominators separately

\frac{7 π}{3 \times 2}

Multiply the numbers

\frac{7 π}{6}

x_{1} = cos (\frac{π}{6}) + i sin (\frac{π}{6}) x_{2} = cos (\frac{7 π}{6}) + i sin (\frac{7 π}{6})

Calculate

x_{1} = \frac{3}{2} + \frac{1}{2} i x_{2} = - \frac{3}{2} - \frac{1}{2} i

x_{1} = \frac{3}{2} + \frac{1}{2} i x_{2} = - \frac{3}{2} - \frac{1}{2} i x^{2} = \frac{1}{2} - \frac{3}{2} i

Solve the equation for x

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Substitute back

x^{2} = \frac{1}{2} - \frac{3}{2} i

Simplify

x = \frac{1}{2} - \frac{3}{2} i

Rewrite the complex number in polar form

More Steps

Evaluate

\frac{1}{2} - \frac{3}{2} i

Determine the modulus and the argument of the complex number

r = (\frac{1}{2})^{2} + (- \frac{3}{2})^{2} θ = arctan \frac{- \frac{3}{2}}{\frac{1}{2}}

Calculate

r = 1 θ = arctan \frac{- \frac{3}{2}}{\frac{1}{2}}

Convert to radians

r = 1 θ = - \frac{π}{3}

Since \frac{1}{2} - \frac{3}{2} i lies in the I V quadrant, add 2 π to get the argument in the I V quadrant

r = 1 θ = - \frac{π}{3} + 2 π

Simplify

r = 1 θ = \frac{5 π}{3}

Substitute the given values into the formula r(cosθ+isinθ)

1 \times (cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3}))

x = 1 \times (cos (\frac{5 π}{3}) + i sin (\frac{5 π}{3}))

Calculate the nth roots of a complex r (cos (θ) + i \times sin (θ),using n z = n r (cos \frac{θ + 2 kπ}{n} + i sin \frac{θ + 2 kπ}{n})

x = 1 \times (cos (\frac{\frac{5 π}{3} + 2 kπ}{2}) + i sin (\frac{\frac{5 π}{3} + 2 kπ}{2}))

Simplify

x = cos (\frac{\frac{5 π}{3} + 2 kπ}{2}) + i sin (\frac{\frac{5 π}{3} + 2 kπ}{2})

Since n = 2,substitute k = 0, 1 into the expression

x_{1} = cos (\frac{\frac{5 π}{3} + 2 \times 0 \times π}{2}) + i sin (\frac{\frac{5 π}{3} + 2 \times 0 \times π}{2}) x_{2} = cos (\frac{\frac{5 π}{3} + 2 \times 1 \times π}{2}) + i sin (\frac{\frac{5 π}{3} + 2 \times 1 \times π}{2})

Calculate

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Evaluate

\frac{\frac{5 π}{3} + 2 \times 0 \times π}{2}

Any expression multiplied by 0 equals 0

\frac{\frac{5 π}{3} + 0}{2}

Removing 0 doesn't change the value,so remove it from the expression

\frac{\frac{5 π}{3}}{2}

Rewrite the expression

\frac{5 π}{3} \times \frac{1}{2}

To multiply the fractions,multiply the numerators and denominators separately

\frac{5 π}{3 \times 2}

Multiply the numbers

\frac{5 π}{6}

x_{1} = cos (\frac{5 π}{6}) + i sin (\frac{5 π}{6}) x_{2} = cos (\frac{\frac{5 π}{3} + 2 \times 1 \times π}{2}) + i sin (\frac{\frac{5 π}{3} + 2 \times 1 \times π}{2})

Calculate

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Evaluate

\frac{\frac{5 π}{3} + 2 \times 1 \times π}{2}

Multiply the terms

\frac{\frac{5 π}{3} + 2 π}{2}

Calculate

\frac{\frac{11 π}{3}}{2}

Rewrite the expression

\frac{11 π}{3} \times \frac{1}{2}

To multiply the fractions,multiply the numerators and denominators separately

\frac{11 π}{3 \times 2}

Multiply the numbers

\frac{11 π}{6}

x_{1} = cos (\frac{5 π}{6}) + i sin (\frac{5 π}{6}) x_{2} = cos (\frac{11 π}{6}) + i sin (\frac{11 π}{6})

Calculate

x_{1} = - \frac{3}{2} + \frac{1}{2} i x_{2} = \frac{3}{2} - \frac{1}{2} i

x_{1} = \frac{3}{2} + \frac{1}{2} i x_{2} = - \frac{3}{2} - \frac{1}{2} i x_{1} = - \frac{3}{2} + \frac{1}{2} i x_{2} = \frac{3}{2} - \frac{1}{2} i

Solution

x \in / R

Show Solution