Algebra Examples

$(x^{2} + 3 x + 19) (x^{2} + 2) = 0$

Step 1

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

$x^{2} + 3 x + 19 = 0$

$x^{2} + 2 = 0$

Step 2

Set $x^{2} + 3 x + 19$ equal to $0$ and solve for $x$ .

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Step 2.1

Set $x^{2} + 3 x + 19$ equal to $0$ .

$x^{2} + 3 x + 19 = 0$

Step 2.2

Solve $x^{2} + 3 x + 19 = 0$ for $x$ .

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Step 2.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 19$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 19)}}{2 \cdot 1}$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Simplify the numerator.

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Step 2.2.3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 19}}{2 \cdot 1}$

Step 2.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 19$ .

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Step 2.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 19}}{2 \cdot 1}$

Step 2.2.3.1.2.2

Multiply $- 4$ by $19$ .

$x = \frac{- 3 \pm \sqrt{9 - 76}}{2 \cdot 1}$

Step 2.2.3.1.3

Subtract $76$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 67}}{2 \cdot 1}$

Step 2.2.3.1.4

Rewrite $- 67$ as $- 1 (67)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 67}}{2 \cdot 1}$

Step 2.2.3.1.5

Rewrite $\sqrt{- 1 (67)}$ as $\sqrt{- 1} \cdot \sqrt{67}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{67}}{2 \cdot 1}$

Step 2.2.3.1.6

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

$x = \frac{- 3 \pm i \sqrt{67}}{2 \cdot 1}$

Step 2.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{67}}{2}$

Step 2.2.4

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{67}}{2}, - \frac{3 + i \sqrt{67}}{2}$

Step 3

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

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Step 3.1

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

$x^{2} + 2 = 0$

Step 3.2

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

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Step 3.2.1

Subtract $2$ from both sides of the equation.

$x^{2} = - 2$

Step 3.2.2

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

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

Step 3.2.3

Simplify $\pm \sqrt{- 2}$ .

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Step 3.2.3.1

Rewrite $- 2$ as $- 1 (2)$ .

$x = \pm \sqrt{- 1 (2)}$

Step 3.2.3.2

Rewrite $\sqrt{- 1 (2)}$ as $\sqrt{- 1} \cdot \sqrt{2}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{2}$

Step 3.2.3.3

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

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

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.

$x = i \sqrt{2}$

Step 3.2.4.2

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

$x = - i \sqrt{2}$

Step 3.2.4.3

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

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

Step 4

The final solution is all the values that make $(x^{2} + 3 x + 19) (x^{2} + 2) = 0$ true.

$x = - \frac{3 - i \sqrt{67}}{2}, - \frac{3 + i \sqrt{67}}{2}, i \sqrt{2}, - i \sqrt{2}$