Algebra Examples

0 = 35 x^{4} - x^{2} + 25

$0 = 35 x^{4} - x^{2} + 25$

Step 1

Rewrite the equation as

35 x^{4} - x^{2} + 25 = 0

$35 x^{4} - x^{2} + 25 = 0$ .

35 x^{4} - x^{2} + 25 = 0

$35 x^{4} - x^{2} + 25 = 0$

Step 2

Substitute

u = x^{2}

$u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

35 u^{2} - u + 25 = 0

$35 u^{2} - u + 25 = 0$

u = x^{2}

$u = x^{2}$

Step 3

Use the quadratic formula to find the solutions.

\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4

Substitute the values

a = 35

$a = 35$ ,

b = - 1

$b = - 1$ , and

c = 25

$c = 25$ into the quadratic formula and solve for

u

$u$ .

\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (35 \cdot 25)}}{2 \cdot 35}

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (35 \cdot 25)}}{2 \cdot 35}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

u = \frac{1 \pm \sqrt{1 - 4 \cdot 35 \cdot 25}}{2 \cdot 35}

$u = \frac{1 \pm \sqrt{1 - 4 \cdot 35 \cdot 25}}{2 \cdot 35}$

Step 5.1.2

Multiply

- 4 \cdot 35 \cdot 25

$- 4 \cdot 35 \cdot 25$ .

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Step 5.1.2.1

Multiply

- 4

$- 4$ by

35

$35$ .

u = \frac{1 \pm \sqrt{1 - 140 \cdot 25}}{2 \cdot 35}

$u = \frac{1 \pm \sqrt{1 - 140 \cdot 25}}{2 \cdot 35}$

Step 5.1.2.2

Multiply

- 140

$- 140$ by

25

$25$ .

u = \frac{1 \pm \sqrt{1 - 3500}}{2 \cdot 35}

$u = \frac{1 \pm \sqrt{1 - 3500}}{2 \cdot 35}$

u = \frac{1 \pm \sqrt{1 - 3500}}{2 \cdot 35}

$u = \frac{1 \pm \sqrt{1 - 3500}}{2 \cdot 35}$

Step 5.1.3

Subtract

3500

$3500$ from

1

$1$ .

u = \frac{1 \pm \sqrt{- 3499}}{2 \cdot 35}

$u = \frac{1 \pm \sqrt{- 3499}}{2 \cdot 35}$

Step 5.1.4

Rewrite

- 3499

$- 3499$ as

- 1 (3499)

$- 1 (3499)$ .

u = \frac{1 \pm \sqrt{- 1 \cdot 3499}}{2 \cdot 35}

$u = \frac{1 \pm \sqrt{- 1 \cdot 3499}}{2 \cdot 35}$

Step 5.1.5

Rewrite

\sqrt{- 1 (3499)}

$\sqrt{- 1 (3499)}$ as

\sqrt{- 1} \cdot \sqrt{3499}

$\sqrt{- 1} \cdot \sqrt{3499}$ .

u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3499}}{2 \cdot 35}

$u = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3499}}{2 \cdot 35}$

Step 5.1.6

Rewrite

$\sqrt{- 1}$ as

$i$ .

$u = \frac{1 \pm i \sqrt{3499}}{2 \cdot 35}$

Step 5.2

Multiply

$2$ by

$35$ .

$u = \frac{1 \pm i \sqrt{3499}}{70}$

Step 6

The final answer is the combination of both solutions.

$u = \frac{1 + i \sqrt{3499}}{70}, \frac{1 - i \sqrt{3499}}{70}$

Step 7

Substitute the real value of

$u = x^{2}$ back into the solved equation.

$x^{2} = \frac{1 + i \sqrt{3499}}{70}$

${(x^{2})}^{1} = \frac{1 - i \sqrt{3499}}{70}$

Step 8

Solve the first equation for

$x$ .

$x^{2} = \frac{1 + i \sqrt{3499}}{70}$

Step 9

Solve the equation for

$x$ .

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Step 9.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1 + i \sqrt{3499}}{70}}$

Step 9.2

Simplify

$\pm \sqrt{\frac{1 + i \sqrt{3499}}{70}}$ .

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Step 9.2.1

Rewrite

$\sqrt{\frac{1 + i \sqrt{3499}}{70}}$ as

$\frac{\sqrt{1 + i \sqrt{3499}}}{\sqrt{70}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}}}{\sqrt{70}}$

Step 9.2.2

Multiply

$\frac{\sqrt{1 + i \sqrt{3499}}}{\sqrt{70}}$ by

$\frac{\sqrt{70}}{\sqrt{70}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}}}{\sqrt{70}} \cdot \frac{\sqrt{70}}{\sqrt{70}}$

Step 9.2.3

Combine and simplify the denominator.

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Step 9.2.3.1

Multiply

$\frac{\sqrt{1 + i \sqrt{3499}}}{\sqrt{70}}$ by

$\frac{\sqrt{70}}{\sqrt{70}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{\sqrt{70} \sqrt{70}}$

Step 9.2.3.2

Raise

$\sqrt{70}$ to the power of

$1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{{\sqrt{70}}^{1} \sqrt{70}}$

Step 9.2.3.3

Raise

$\sqrt{70}$ to the power of

$1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{{\sqrt{70}}^{1} {\sqrt{70}}^{1}}$

Step 9.2.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{{\sqrt{70}}^{1 + 1}}$

Step 9.2.3.5

Add

$1$ and

$1$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{{\sqrt{70}}^{2}}$

Step 9.2.3.6

Rewrite

${\sqrt{70}}^{2}$ as

$70$ .

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Step 9.2.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{70}$ as

$70^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{{(70^{\frac{1}{2}})}^{2}}$

Step 9.2.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{70^{\frac{1}{2} \cdot 2}}$

Step 9.2.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{70^{\frac{2}{2}}}$

Step 9.2.3.6.4

Cancel the common factor of

$2$ .

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Step 9.2.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{70^{\frac{2}{2}}}$

Step 9.2.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{70^{1}}$

Step 9.2.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{1 + i \sqrt{3499}} \sqrt{70}}{70}$

Step 9.2.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(1 + i \sqrt{3499}) \cdot 70}}{70}$

Step 9.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 9.3.1

First, use the positive value of the

$\pm$ to find the first solution.

$x = \frac{\sqrt{(1 + i \sqrt{3499}) \cdot 70}}{70}$

Step 9.3.2

Next, use the negative value of the

$\pm$ to find the second solution.

$x = - \frac{\sqrt{(1 + i \sqrt{3499}) \cdot 70}}{70}$

Step 9.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{(1 + i \sqrt{3499}) \cdot 70}}{70}, - \frac{\sqrt{(1 + i \sqrt{3499}) \cdot 70}}{70}$

Step 10

Solve the second equation for

$x$ .

${(x^{2})}^{1} = \frac{1 - i \sqrt{3499}}{70}$

Step 11

Solve the equation for

$x$ .

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Step 11.1

Remove parentheses.

$x^{2} = \frac{1 - i \sqrt{3499}}{70}$

Step 11.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1 - i \sqrt{3499}}{70}}$

Step 11.3

Simplify

$\pm \sqrt{\frac{1 - i \sqrt{3499}}{70}}$ .

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Step 11.3.1

Rewrite

$\sqrt{\frac{1 - i \sqrt{3499}}{70}}$ as

$\frac{\sqrt{1 - i \sqrt{3499}}}{\sqrt{70}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}}}{\sqrt{70}}$

Step 11.3.2

Multiply

$\frac{\sqrt{1 - i \sqrt{3499}}}{\sqrt{70}}$ by

$\frac{\sqrt{70}}{\sqrt{70}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}}}{\sqrt{70}} \cdot \frac{\sqrt{70}}{\sqrt{70}}$

Step 11.3.3

Combine and simplify the denominator.

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Step 11.3.3.1

Multiply

$\frac{\sqrt{1 - i \sqrt{3499}}}{\sqrt{70}}$ by

$\frac{\sqrt{70}}{\sqrt{70}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{\sqrt{70} \sqrt{70}}$

Step 11.3.3.2

Raise

$\sqrt{70}$ to the power of

$1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{{\sqrt{70}}^{1} \sqrt{70}}$

Step 11.3.3.3

Raise

$\sqrt{70}$ to the power of

$1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{{\sqrt{70}}^{1} {\sqrt{70}}^{1}}$

Step 11.3.3.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{{\sqrt{70}}^{1 + 1}}$

Step 11.3.3.5

Add

$1$ and

$1$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{{\sqrt{70}}^{2}}$

Step 11.3.3.6

Rewrite

${\sqrt{70}}^{2}$ as

$70$ .

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Step 11.3.3.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{70}$ as

$70^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{{(70^{\frac{1}{2}})}^{2}}$

Step 11.3.3.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{70^{\frac{1}{2} \cdot 2}}$

Step 11.3.3.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{70^{\frac{2}{2}}}$

Step 11.3.3.6.4

Cancel the common factor of

$2$ .

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Step 11.3.3.6.4.1

Cancel the common factor.

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{70^{\frac{2}{2}}}$

Step 11.3.3.6.4.2

Rewrite the expression.

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{70^{1}}$

Step 11.3.3.6.5

Evaluate the exponent.

$x = \pm \frac{\sqrt{1 - i \sqrt{3499}} \sqrt{70}}{70}$

Step 11.3.4

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(1 - i \sqrt{3499}) \cdot 70}}{70}$

Step 11.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 11.4.1

First, use the positive value of the

$\pm$ to find the first solution.

$x = \frac{\sqrt{(1 - i \sqrt{3499}) \cdot 70}}{70}$

Step 11.4.2

Next, use the negative value of the

$\pm$ to find the second solution.

$x = - \frac{\sqrt{(1 - i \sqrt{3499}) \cdot 70}}{70}$

Step 11.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{(1 - i \sqrt{3499}) \cdot 70}}{70}, - \frac{\sqrt{(1 - i \sqrt{3499}) \cdot 70}}{70}$

Step 12

The solution to

$35 x^{4} - x^{2} + 25 = 0$ is

$x = \frac{\sqrt{(1 + i \sqrt{3499}) \cdot 70}}{70}, - \frac{\sqrt{(1 + i \sqrt{3499}) \cdot 70}}{70}, \frac{\sqrt{(1 - i \sqrt{3499}) \cdot 70}}{70}, - \frac{\sqrt{(1 - i \sqrt{3499}) \cdot 70}}{70}$ .

Step 13