Algebra Examples

$x^{4} + 4 x^{3} + 7 x^{2} + 16 x + 12 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Regroup terms.

$4 x^{3} + 16 x + x^{4} + 7 x^{2} + 12 = 0$

Step 1.2

Factor $4 x$ out of $4 x^{3} + 16 x$ .

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Step 1.2.1

Factor $4 x$ out of $4 x^{3}$ .

$4 x (x^{2}) + 16 x + x^{4} + 7 x^{2} + 12 = 0$

Step 1.2.2

Factor $4 x$ out of $16 x$ .

$4 x (x^{2}) + 4 x (4) + x^{4} + 7 x^{2} + 12 = 0$

Step 1.2.3

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

$4 x (x^{2} + 4) + x^{4} + 7 x^{2} + 12 = 0$

Step 1.3

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

$4 x (x^{2} + 4) + {(x^{2})}^{2} + 7 x^{2} + 12 = 0$

Step 1.4

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

$4 x (x^{2} + 4) + u^{2} + 7 u + 12 = 0$

Step 1.5

Factor $u^{2} + 7 u + 12$ using the AC method.

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Step 1.5.1

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 $12$ and whose sum is $7$ .

$3, 4$

Step 1.5.2

Write the factored form using these integers.

$4 x (x^{2} + 4) + (u + 3) (u + 4) = 0$

Step 1.6

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

$4 x (x^{2} + 4) + (x^{2} + 3) (x^{2} + 4) = 0$

Step 1.7

Factor $x^{2} + 4$ out of $4 x (x^{2} + 4) + (x^{2} + 3) (x^{2} + 4)$ .

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Step 1.7.1

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

$(x^{2} + 4) (4 x) + (x^{2} + 3) (x^{2} + 4) = 0$

Step 1.7.2

Factor $x^{2} + 4$ out of $(x^{2} + 3) (x^{2} + 4)$ .

$(x^{2} + 4) (4 x) + (x^{2} + 4) (x^{2} + 3) = 0$

Step 1.7.3

Factor $x^{2} + 4$ out of $(x^{2} + 4) (4 x) + (x^{2} + 4) (x^{2} + 3)$ .

$(x^{2} + 4) (4 x + x^{2} + 3) = 0$

Step 1.8

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

$(x^{2} + 4) (4 u + u^{2} + 3) = 0$

Step 1.9

Factor $4 u + u^{2} + 3$ using the AC method.

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Step 1.9.1

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 $3$ and whose sum is $4$ .

$1, 3$

Step 1.9.2

Write the factored form using these integers.

$(x^{2} + 4) ((u + 1) (u + 3)) = 0$

Step 1.10

Factor.

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Step 1.10.1

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

$(x^{2} + 4) ((x + 1) (x + 3)) = 0$

Step 1.10.2

Remove unnecessary parentheses.

$(x^{2} + 4) (x + 1) (x + 3) = 0$

Step 2

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

$x^{2} + 4 = 0$

$x + 1 = 0$

$x + 3 = 0$

Step 3

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

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Step 3.1

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

$x^{2} + 4 = 0$

Step 3.2

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

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Step 3.2.1

Subtract $4$ from both sides of the equation.

$x^{2} = - 4$

Step 3.2.2

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

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

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

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

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

Step 3.2.3.3

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

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

Step 3.2.3.4

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

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

Step 3.2.3.5

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

$x = \pm i \cdot 2$

Step 3.2.3.6

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

$x = \pm 2 i$

Step 3.2.4

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

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Step 3.2.4.1

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

$x = 2 i$

Step 3.2.4.2

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

$x = - 2 i$

Step 3.2.4.3

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 $x$ .

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Step 4.1

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

$x + 1 = 0$

Step 4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5

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

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Step 5.1

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

$x + 3 = 0$

Step 5.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 6

The final solution is all the values that make $(x^{2} + 4) (x + 1) (x + 3) = 0$ true.

$x = 2 i, - 2 i, - 1, - 3$

Step 7