Precalculus Examples

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

Step 1

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

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

Step 2

Rewrite the equation as $| x | = x^{2} + x - 3$ .

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

Step 3

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

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

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.

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

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.

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

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.

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

Step 4.3.2

Combine the opposite terms in $x^{2} + x - 3 - x$ .

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Step 4.3.2.1

Subtract $x$ from $x$ .

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

Step 4.3.2.2

Add $x^{2}$ and $0$ .

$x^{2} - 3 = 0$

Step 4.4

Add $3$ to both sides of the equation.

$x^{2} = 3$

Step 4.5

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

$x = \pm \sqrt{3}$

Step 4.6

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

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Step 4.6.1

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

$x = \sqrt{3}$

Step 4.6.2

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

$x = - \sqrt{3}$

Step 4.6.3

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

$x = \sqrt{3}, - \sqrt{3}$

Step 4.7

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

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

Step 4.8

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

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

Step 4.9

Simplify $- (x^{2} + x - 3)$ .

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Step 4.9.1

Rewrite.

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

Step 4.9.2

Simplify by adding zeros.

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

Step 4.9.3

Apply the distributive property.

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

Step 4.9.4

Multiply $- 1$ by $- 3$ .

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

Step 4.10

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

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Step 4.10.1

Subtract $x$ from both sides of the equation.

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

Step 4.10.2

Subtract $x$ from $- x$ .

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

Step 4.11

Factor the left side of the equation.

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Step 4.11.1

Factor $- 1$ out of $- x^{2} - 2 x + 3$ .

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Step 4.11.1.1

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

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

Step 4.11.1.2

Factor $- 1$ out of $- 2 x$ .

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

Step 4.11.1.3

Rewrite $3$ as $- 1 (- 3)$ .

$- (x^{2}) - (2 x) - 1 \cdot - 3 = 0$

Step 4.11.1.4

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

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

Step 4.11.1.5

Factor $- 1$ out of $- (x^{2} + 2 x) - 1 (- 3)$ .

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

Step 4.11.2

Factor.

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Step 4.11.2.1

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

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Step 4.11.2.1.1

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 $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 4.11.2.1.2

Write the factored form using these integers.

$- ((x - 1) (x + 3)) = 0$

Step 4.11.2.2

Remove unnecessary parentheses.

$- (x - 1) (x + 3) = 0$

Step 4.12

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

$x - 1 = 0$

$x + 3 = 0$

Step 4.13

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

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Step 4.13.1

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

$x - 1 = 0$

Step 4.13.2

Add $1$ to both sides of the equation.

$x = 1$

Step 4.14

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

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Step 4.14.1

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

$x + 3 = 0$

Step 4.14.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 4.15

The final solution is all the values that make $- (x - 1) (x + 3) = 0$ true.

$x = 1, - 3$

Step 4.16

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

$x = \sqrt{3}, - \sqrt{3}, 1, - 3$

Step 5

Exclude the solutions that do not make $| x | = x^{2} + x - 3$ true.

$x = \sqrt{3}, - 3$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{3}, - 3$

Decimal Form:

$x = 1.73205080 \dots, - 3$