Solve for x x^2(x^2+31)=180

$x^{2} (x^{2} + 31) = 180$

Simplify $x^{2} (x^{2} + 31)$ .

Apply the distributive property.

$x^{2} x^{2} + x^{2} \cdot 31 = 180$

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x^{2 + 2} + x^{2} \cdot 31 = 180$

Add $2$ and $2$ .

$x^{4} + x^{2} \cdot 31 = 180$

Move $31$ to the left of $x^{2}$ .

$x^{4} + 31 x^{2} = 180$

Subtract $180$ from both sides of the equation.

$x^{4} + 31 x^{2} - 180 = 0$

Factor the left side of the equation.

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

${(x^{2})}^{2} + 31 x^{2} - 180 = 0$

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

$u^{2} + 31 u - 180 = 0$

Factor $u^{2} + 31 u - 180$ 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 $- 180$ and whose sum is $31$ .

$- 5, 36$

Write the factored form using these integers.

$(u - 5) (u + 36) = 0$

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

$(x^{2} - 5) (x^{2} + 36) = 0$

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

$x^{2} - 5 = 0$

$x^{2} + 36 = 0$

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 $x^{2} + 36$ equal to $0$ and solve for $x$ .

Set $x^{2} + 36$ equal to $0$ .

$x^{2} + 36 = 0$

Solve $x^{2} + 36 = 0$ for $x$ .

Subtract $36$ from both sides of the equation.

$x^{2} = - 36$

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

$x = \pm \sqrt{- 36}$

Simplify $\pm \sqrt{- 36}$ .

Rewrite $- 36$ as $- 1 (36)$ .

$x = \pm \sqrt{- 1 (36)}$

Rewrite $\sqrt{- 1 (36)}$ as $\sqrt{- 1} \cdot \sqrt{36}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{36}$

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

$x = \pm i \cdot \sqrt{36}$

Rewrite $36$ as $6^{2}$ .

$x = \pm i \cdot \sqrt{6^{2}}$

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

$x = \pm i \cdot 6$

Move $6$ to the left of $i$ .

$x = \pm 6 i$

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.

$x = 6 i$

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

$x = - 6 i$

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

$x = 6 i, - 6 i$

The final solution is all the values that make $(x^{2} - 5) (x^{2} + 36) = 0$ true.

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

Solve for x x^2(x^2+31)=180

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