Finite Math Examples

$x^{4} - 7 = 6 x^{2}$

Step 1

Subtract $6 x^{2}$ from both sides of the equation.

$x^{4} - 7 - 6 x^{2} = 0$

Step 2

Factor $x^{4} - 7 - 6 x^{2}$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 2.2

Write the factored form using these integers.

$(x^{2} - 7) (x^{2} + 1) = 0$

Step 3

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

$x^{2} - 7 = 0$

$x^{2} + 1 = 0$

Step 4

Set $x^{2} - 7$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{2} - 7$ equal to $0$ .

$x^{2} - 7 = 0$

Step 4.2

Solve $x^{2} - 7 = 0$ for $x$ .

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Step 4.2.1

Add $7$ to both sides of the equation.

$x^{2} = 7$

Step 4.2.2

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

$x = \pm \sqrt{7}$

Step 4.2.3

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

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Step 4.2.3.1

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

$x = \sqrt{7}$

Step 4.2.3.2

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

$x = - \sqrt{7}$

Step 4.2.3.3

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

$x = \sqrt{7}, - \sqrt{7}$

Step 5

Set $x^{2} + 1$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + 1$ equal to $0$ .

$x^{2} + 1 = 0$

Step 5.2

Solve $x^{2} + 1 = 0$ for $x$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$x^{2} = - 1$

Step 5.2.2

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

$x = \pm \sqrt{- 1}$

Step 5.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i$

Step 5.2.4

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

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Step 5.2.4.1

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

$x = i$

Step 5.2.4.2

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

$x = - i$

Step 5.2.4.3

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

$x = i, - i$

Step 6

The final solution is all the values that make $(x^{2} - 7) (x^{2} + 1) = 0$ true.

$x = \sqrt{7}, - \sqrt{7}, i, - i$