Algebra Examples

$| x + 3 | - 1 = {(x + 2)}^{2}$

Step 1

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

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Step 1.1

Rewrite.

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

Step 1.2

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

$| x + 3 | - 1 = (x + 2) (x + 2)$

Step 1.3

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 1.3.1

Apply the distributive property.

$| x + 3 | - 1 = x (x + 2) + 2 (x + 2)$

Step 1.3.2

Apply the distributive property.

$| x + 3 | - 1 = x \cdot x + x \cdot 2 + 2 (x + 2)$

Step 1.3.3

Apply the distributive property.

$| x + 3 | - 1 = x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2$

Step 1.4

Simplify and combine like terms.

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Step 1.4.1

Simplify each term.

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Step 1.4.1.1

Multiply $x$ by $x$ .

$| x + 3 | - 1 = x^{2} + x \cdot 2 + 2 x + 2 \cdot 2$

Step 1.4.1.2

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

$| x + 3 | - 1 = x^{2} + 2 \cdot x + 2 x + 2 \cdot 2$

Step 1.4.1.3

Multiply $2$ by $2$ .

$| x + 3 | - 1 = x^{2} + 2 x + 2 x + 4$

Step 1.4.2

Add $2 x$ and $2 x$ .

$| x + 3 | - 1 = x^{2} + 4 x + 4$

Step 2

Move all terms not containing $| x + 3 |$ to the right side of the equation.

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Step 2.1

Add $1$ to both sides of the equation.

$| x + 3 | = x^{2} + 4 x + 4 + 1$

Step 2.2

Add $4$ and $1$ .

$| x + 3 | = x^{2} + 4 x + 5$

Step 3

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x + 3 = \pm (x^{2} + 4 x + 5)$

Step 4

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

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Step 4.1

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

$x + 3 = x^{2} + 4 x + 5$

Step 4.2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 4 x + 5 = x + 3$

Step 4.3

Move all terms containing $x$ to the left side of the equation.

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Step 4.3.1

Subtract $x$ from both sides of the equation.

$x^{2} + 4 x + 5 - x = 3$

Step 4.3.2

Subtract $x$ from $4 x$ .

$x^{2} + 3 x + 5 = 3$

Step 4.4

Subtract $3$ from both sides of the equation.

$x^{2} + 3 x + 5 - 3 = 0$

Step 4.5

Subtract $3$ from $5$ .

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

Step 4.6

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

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Step 4.6.1

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 $2$ and whose sum is $3$ .

$1, 2$

Step 4.6.2

Write the factored form using these integers.

$(x + 1) (x + 2) = 0$

Step 4.7

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

$x + 1 = 0$

$x + 2 = 0$

Step 4.8

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

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Step 4.8.1

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

$x + 1 = 0$

Step 4.8.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.9

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

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Step 4.9.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 4.9.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4.10

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

$x = - 1, - 2$

Step 4.11

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

$x + 3 = - (x^{2} + 4 x + 5)$

Step 4.12

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- (x^{2} + 4 x + 5) = x + 3$

Step 4.13

Simplify $- (x^{2} + 4 x + 5)$ .

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Step 4.13.1

Rewrite.

$0 + 0 - (x^{2} + 4 x + 5) = x + 3$

Step 4.13.2

Simplify by adding zeros.

$- (x^{2} + 4 x + 5) = x + 3$

Step 4.13.3

Apply the distributive property.

$- x^{2} - (4 x) - 1 \cdot 5 = x + 3$

Step 4.13.4

Simplify.

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Step 4.13.4.1

Multiply $4$ by $- 1$ .

$- x^{2} - 4 x - 1 \cdot 5 = x + 3$

Step 4.13.4.2

Multiply $- 1$ by $5$ .

$- x^{2} - 4 x - 5 = x + 3$

Step 4.14

Move all terms containing $x$ to the left side of the equation.

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Step 4.14.1

Subtract $x$ from both sides of the equation.

$- x^{2} - 4 x - 5 - x = 3$

Step 4.14.2

Subtract $x$ from $- 4 x$ .

$- x^{2} - 5 x - 5 = 3$

Step 4.15

Move all terms to the left side of the equation and simplify.

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Step 4.15.1

Subtract $3$ from both sides of the equation.

$- x^{2} - 5 x - 5 - 3 = 0$

Step 4.15.2

Subtract $3$ from $- 5$ .

$- x^{2} - 5 x - 8 = 0$

Step 4.16

Use the quadratic formula to find the solutions.

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

Step 4.17

Substitute the values $a = - 1$ , $b = - 5$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (- 1 \cdot - 8)}}{2 \cdot - 1}$

Step 4.18

Simplify.

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Step 4.18.1

Simplify the numerator.

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Step 4.18.1.1

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot - 1 \cdot - 8}}{2 \cdot - 1}$

Step 4.18.1.2

Multiply $- 4 \cdot - 1 \cdot - 8$ .

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Step 4.18.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{5 \pm \sqrt{25 + 4 \cdot - 8}}{2 \cdot - 1}$

Step 4.18.1.2.2

Multiply $4$ by $- 8$ .

$x = \frac{5 \pm \sqrt{25 - 32}}{2 \cdot - 1}$

Step 4.18.1.3

Subtract $32$ from $25$ .

$x = \frac{5 \pm \sqrt{- 7}}{2 \cdot - 1}$

Step 4.18.1.4

Rewrite $- 7$ as $- 1 (7)$ .

$x = \frac{5 \pm \sqrt{- 1 \cdot 7}}{2 \cdot - 1}$

Step 4.18.1.5

Rewrite $\sqrt{- 1 (7)}$ as $\sqrt{- 1} \cdot \sqrt{7}$ .

$x = \frac{5 \pm \sqrt{- 1} \cdot \sqrt{7}}{2 \cdot - 1}$

Step 4.18.1.6

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

$x = \frac{5 \pm i \sqrt{7}}{2 \cdot - 1}$

Step 4.18.2

Multiply $2$ by $- 1$ .

$x = \frac{5 \pm i \sqrt{7}}{- 2}$

Step 4.18.3

Move the negative in front of the fraction.

$x = - \frac{5 \pm i \sqrt{7}}{2}$

Step 4.19

The final answer is the combination of both solutions.

$x = - \frac{5 + i \sqrt{7}}{2}, - \frac{5 - i \sqrt{7}}{2}$

Step 4.20

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

$x = - 1, - 2, - \frac{5 + i \sqrt{7}}{2}, - \frac{5 - i \sqrt{7}}{2}$