Algebra Examples

$x^{2} = 2 - | x |$

Step 1

Rewrite the equation as $2 - | x | = x^{2}$ .

$2 - | x | = x^{2}$

Step 2

Subtract $2$ from both sides of the equation.

$- | x | = x^{2} - 2$

Step 3

Divide each term in $- | x | = x^{2} - 2$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- | x | = x^{2} - 2$ by $- 1$ .

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

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2

Divide $| x |$ by $1$ .

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

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Move the negative one from the denominator of $\frac{x^{2}}{- 1}$ .

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

Step 3.3.1.2

Rewrite $- 1 \cdot x^{2}$ as $- x^{2}$ .

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

Step 3.3.1.3

Divide $- 2$ by $- 1$ .

$| x | = - x^{2} + 2$

Step 4

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

$x = \pm (- x^{2} + 2)$

Step 5

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

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Step 5.1

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

$x = - x^{2} + 2$

Step 5.2

Add $x^{2}$ to both sides of the equation.

$x + x^{2} = 2$

Step 5.3

Subtract $2$ from both sides of the equation.

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

Step 5.4

Factor the left side of the equation.

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Step 5.4.1

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

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

Step 5.4.2

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

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Step 5.4.2.1

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 $1$ .

$- 1, 2$

Step 5.4.2.2

Write the factored form using these integers.

$(u - 1) (u + 2) = 0$

Step 5.4.3

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

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

Step 5.5

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 5.6

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

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Step 5.6.1

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

$x - 1 = 0$

Step 5.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 5.7

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

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Step 5.7.1

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

$x + 2 = 0$

Step 5.7.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 5.8

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

$x = 1, - 2$

Step 5.9

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

$x = - (- x^{2} + 2)$

Step 5.10

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

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Step 5.10.1

Apply the distributive property.

$x = - - x^{2} - 1 \cdot 2$

Step 5.10.2

Multiply $- - x^{2}$ .

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Step 5.10.2.1

Multiply $- 1$ by $- 1$ .

$x = 1 x^{2} - 1 \cdot 2$

Step 5.10.2.2

Multiply $x^{2}$ by $1$ .

$x = x^{2} - 1 \cdot 2$

Step 5.10.3

Multiply $- 1$ by $2$ .

$x = x^{2} - 2$

Step 5.11

Subtract $x^{2}$ from both sides of the equation.

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

Step 5.12

Add $2$ to both sides of the equation.

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

Step 5.13

Factor the left side of the equation.

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Step 5.13.1

Factor $- 1$ out of $x - x^{2} + 2$ .

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Step 5.13.1.1

Reorder $x$ and $- x^{2}$ .

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

Step 5.13.1.2

Factor $- 1$ out of $- x^{2}$ .

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

Step 5.13.1.3

Factor $- 1$ out of $x$ .

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

Step 5.13.1.4

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

$- (x^{2}) - 1 (- x) - 1 \cdot - 2 = 0$

Step 5.13.1.5

Factor $- 1$ out of $- (x^{2}) - 1 (- x)$ .

$- (x^{2} - x) - 1 \cdot - 2 = 0$

Step 5.13.1.6

Factor $- 1$ out of $- (x^{2} - x) - 1 (- 2)$ .

$- (x^{2} - x - 2) = 0$

Step 5.13.2

Factor.

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Step 5.13.2.1

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

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Step 5.13.2.1.1

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 $- 1$ .

$- 2, 1$

Step 5.13.2.1.2

Write the factored form using these integers.

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

Step 5.13.2.2

Remove unnecessary parentheses.

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

Step 5.14

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

$x - 2 = 0$

$x + 1 = 0$

Step 5.15

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

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Step 5.15.1

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

$x - 2 = 0$

Step 5.15.2

Add $2$ to both sides of the equation.

$x = 2$

Step 5.16

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

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Step 5.16.1

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

$x + 1 = 0$

Step 5.16.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 5.17

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

$x = 2, - 1$

Step 5.18

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

$x = 1, - 2, 2, - 1$

Step 6

Exclude the solutions that do not make $x^{2} = 2 - | x |$ true.

$x = 1, - 1$