Find the Roots (Zeros) x^4+6x^3+9x^2+24x+20=0

$x^{4} + 6 x^{3} + 9 x^{2} + 24 x + 20 = 0$

Step 1

Factor the left side of the equation.

Regroup terms.

$6 x^{3} + 24 x + x^{4} + 9 x^{2} + 20 = 0$

Factor $6 x$ out of $6 x^{3} + 24 x$ .

Factor $6 x$ out of $6 x^{3}$ .

$6 x (x^{2}) + 24 x + x^{4} + 9 x^{2} + 20 = 0$

Factor $6 x$ out of $24 x$ .

$6 x (x^{2}) + 6 x (4) + x^{4} + 9 x^{2} + 20 = 0$

Factor $6 x$ out of $6 x (x^{2}) + 6 x (4)$ .

$6 x (x^{2} + 4) + x^{4} + 9 x^{2} + 20 = 0$

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

$6 x (x^{2} + 4) + {(x^{2})}^{2} + 9 x^{2} + 20 = 0$

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

$6 x (x^{2} + 4) + u^{2} + 9 u + 20 = 0$

Factor $u^{2} + 9 u + 20$ 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 $20$ and whose sum is $9$ .

$4, 5$

Write the factored form using these integers.

$6 x (x^{2} + 4) + (u + 4) (u + 5) = 0$

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

$6 x (x^{2} + 4) + (x^{2} + 4) (x^{2} + 5) = 0$

Factor $x^{2} + 4$ out of $6 x (x^{2} + 4) + (x^{2} + 4) (x^{2} + 5)$ .

Factor $x^{2} + 4$ out of $6 x (x^{2} + 4)$ .

$(x^{2} + 4) (6 x) + (x^{2} + 4) (x^{2} + 5) = 0$

Factor $x^{2} + 4$ out of $(x^{2} + 4) (6 x) + (x^{2} + 4) (x^{2} + 5)$ .

$(x^{2} + 4) (6 x + x^{2} + 5) = 0$

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$(x^{2} + 4) (6 u + u^{2} + 5) = 0$

Factor $6 u + u^{2} + 5$ 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 $5$ and whose sum is $6$ .

$1, 5$

Write the factored form using these integers.

$(x^{2} + 4) ((u + 1) (u + 5)) = 0$

Factor.

Replace all occurrences of $u$ with $x$ .

$(x^{2} + 4) ((x + 1) (x + 5)) = 0$

Remove unnecessary parentheses.

$(x^{2} + 4) (x + 1) (x + 5) = 0$

Step 2

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

$x^{2} + 4 = 0$

$x + 1 = 0$

$x + 5 = 0$

Step 3

Set $x^{2} + 4$ equal to $0$ and solve for $x$ .

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

$x^{2} + 4 = 0$

Solve $x^{2} + 4 = 0$ for $x$ .

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

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

$x = \pm \sqrt{- 4}$

Simplify $\pm \sqrt{- 4}$ .

Rewrite $- 4$ as $- 1 (4)$ .

$x = \pm \sqrt{- 1 (4)}$

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{4}$

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

$x = \pm i \cdot \sqrt{4}$

Rewrite $4$ as $2^{2}$ .

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

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

$x = \pm i \cdot 2$

Move $2$ to the left of $i$ .

$x = \pm 2 i$

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.

$x = 2 i$

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

$x = - 2 i$

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

$x = 2 i, - 2 i$

Step 4

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

Set $x + 5$ equal to $0$ and solve for $x$ .

Set $x + 5$ equal to $0$ .

$x + 5 = 0$

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 6

The final solution is all the values that make $(x^{2} + 4) (x + 1) (x + 5) = 0$ true.

$x = 2 i, - 2 i, - 1, - 5$

Step 7

Find the Roots (Zeros) x^4+6x^3+9x^2+24x+20=0

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