Finite Math Examples

$x^{7} + x^{2} = 6 x^{5} + 6$

Step 1

Subtract $6 x^{5}$ from both sides of the equation.

$x^{7} + x^{2} - 6 x^{5} = 6$

Step 2

Subtract $6$ from both sides of the equation.

$x^{7} + x^{2} - 6 x^{5} - 6 = 0$

Step 3

Factor the left side of the equation.

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Step 3.1

Reorder terms.

$x^{7} - 6 x^{5} + x^{2} - 6 = 0$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$(x^{7} - 6 x^{5}) + x^{2} - 6 = 0$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$x^{5} (x^{2} - 6) + 1 (x^{2} - 6) = 0$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $x^{2} - 6$ .

$(x^{2} - 6) (x^{5} + 1) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x^{2} - 6 = 0$

$x^{5} + 1 = 0$

Step 5

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} - 6$ equal to $0$ .

$x^{2} - 6 = 0$

Step 5.2

Solve $x^{2} - 6 = 0$ for $x$ .

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Step 5.2.1

Add $6$ to both sides of the equation.

$x^{2} = 6$

Step 5.2.2

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

$x = \pm \sqrt{6}$

Step 5.2.3

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

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Step 5.2.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{6}$

Step 5.2.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{6}$

Step 5.2.3.3

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

$x = \sqrt{6}, - \sqrt{6}$

Step 6

Set $x^{5} + 1$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{5} + 1$ equal to $0$ .

$x^{5} + 1 = 0$

Step 6.2

Solve $x^{5} + 1 = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$x^{5} = - 1$

Step 6.2.2

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

$x = \sqrt[5]{- 1}$

Step 6.2.3

Simplify $\sqrt[5]{- 1}$ .

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Step 6.2.3.1

Rewrite $- 1$ as ${(- 1)}^{5}$ .

$x = \sqrt[5]{{(- 1)}^{5}}$

Step 6.2.3.2

Pull terms out from under the radical, assuming real numbers.

$x = - 1$

Step 7

The final solution is all the values that make $(x^{2} - 6) (x^{5} + 1) = 0$ true.

$x = \sqrt{6}, - \sqrt{6}, - 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{6}, - \sqrt{6}, - 1$

Decimal Form:

$x = 2.44948974 \dots, - 2.44948974 \dots, - 1$