Find the Roots (Zeros) f(x)=x^6+x^4-30x^2

$f (x) = x^{6} + x^{4} - 30 x^{2}$

Set $x^{6} + x^{4} - 30 x^{2}$ equal to $0$ .

$x^{6} + x^{4} - 30 x^{2} = 0$

Solve for $x$ .

Factor the left side of the equation.

Factor $x^{2}$ out of $x^{6} + x^{4} - 30 x^{2}$ .

Factor $x^{2}$ out of $x^{6}$ .

$x^{2} x^{4} + x^{4} - 30 x^{2} = 0$

Factor $x^{2}$ out of $x^{4}$ .

$x^{2} x^{4} + x^{2} x^{2} - 30 x^{2} = 0$

Factor $x^{2}$ out of $- 30 x^{2}$ .

$x^{2} x^{4} + x^{2} x^{2} + x^{2} \cdot - 30 = 0$

Factor $x^{2}$ out of $x^{2} x^{4} + x^{2} x^{2}$ .

$x^{2} (x^{4} + x^{2}) + x^{2} \cdot - 30 = 0$

Factor $x^{2}$ out of $x^{2} (x^{4} + x^{2}) + x^{2} \cdot - 30$ .

$x^{2} (x^{4} + x^{2} - 30) = 0$

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$x^{2} ({(x^{2})}^{2} + x^{2} - 30) = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$x^{2} (u^{2} + u - 30) = 0$

Factor $u^{2} + u - 30$ 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 $- 30$ and whose sum is $1$ .

$- 5, 6$

Write the factored form using these integers.

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

Factor.

Replace all occurrences of $u$ with $x^{2}$ .

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

Remove unnecessary parentheses.

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

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

$x^{2} = 0$

$x^{2} - 5 = 0$

$x^{2} + 6 = 0$

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

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

$x^{2} = 0$

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

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Solve $x^{2} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$x^{2} = 5$

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

$x = \pm \sqrt{5}$

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.

$x = \sqrt{5}$

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

$x = - \sqrt{5}$

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

$x = \sqrt{5}, - \sqrt{5}$

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

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

$x^{2} + 6 = 0$

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

Subtract $6$ from both sides of the equation.

$x^{2} = - 6$

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

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

Simplify $\pm \sqrt{- 6}$ .

Rewrite $- 6$ as $- 1 (6)$ .

$x = \pm \sqrt{- 1 (6)}$

Rewrite $\sqrt{- 1 (6)}$ as $\sqrt{- 1} \cdot \sqrt{6}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{6}$

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

$x = \pm i \sqrt{6}$

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.

$x = i \sqrt{6}$

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

$x = - i \sqrt{6}$

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

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

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

$x = 0, \sqrt{5}, - \sqrt{5}, i \sqrt{6}, - i \sqrt{6}$

Find the Roots (Zeros) f(x)=x^6+x^4-30x^2

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