Solve by Factoring x^4+18=11x^2

$x^{4} + 18 = 11 x^{2}$

Step 1

Subtract $11 x^{2}$ from both sides of the equation.

$x^{4} + 18 - 11 x^{2} = 0$

Step 2

Factor $x^{4} + 18 - 11 x^{2}$ using the AC method.

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 $18$ and whose sum is $- 11$ .

$- 9, - 2$

Write the factored form using these integers.

$(x^{2} - 9) (x^{2} - 2) = 0$

Step 3

Rewrite $9$ as $3^{2}$ .

$(x^{2} - 3^{2}) (x^{2} - 2) = 0$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$(x + 3) (x - 3) (x^{2} - 2) = 0$

Step 5

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

$x + 3 = 0$

$x - 3 = 0$

$x^{2} - 2 = 0$

Step 6

Set $x + 3$ equal to $0$ and solve for $x$ .

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 7

Set $x - 3$ equal to $0$ and solve for $x$ .

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

Step 8

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

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

$x^{2} - 2 = 0$

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

Add $2$ to both sides of the equation.

$x^{2} = 2$

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

$x = \pm \sqrt{2}$

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

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

$x = \sqrt{2}$

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

$x = - \sqrt{2}$

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

$x = \sqrt{2}, - \sqrt{2}$

Step 9

The final solution is all the values that make $(x + 3) (x - 3) (x^{2} - 2) = 0$ true.

$x = - 3, 3, \sqrt{2}, - \sqrt{2}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$x = - 3, 3, \sqrt{2}, - \sqrt{2}$

Decimal Form:

$x = - 3, 3, 1.41421356 \dots, - 1.41421356 \dots$

Solve by Factoring x^4+18=11x^2

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