Algebra Examples

$x^{2} + 3 x + 7 = - | 2 x | + 12$

Step 1

Rewrite the equation as $- | 2 x | + 12 = x^{2} + 3 x + 7$ .

$- | 2 x | + 12 = x^{2} + 3 x + 7$

Step 2

Move all terms not containing $| 2 x |$ to the right side of the equation.

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Step 2.1

Subtract $12$ from both sides of the equation.

$- | 2 x | = x^{2} + 3 x + 7 - 12$

Step 2.2

Subtract $12$ from $7$ .

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

Step 3

Divide each term in $- | 2 x | = x^{2} + 3 x - 5$ by $- 1$ and simplify.

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Step 3.1

Divide each term in $- | 2 x | = x^{2} + 3 x - 5$ by $- 1$ .

$\frac{- | 2 x |}{- 1} = \frac{x^{2}}{- 1} + \frac{3 x}{- 1} + \frac{- 5}{- 1}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Dividing two negative values results in a positive value.

$\frac{| 2 x |}{1} = \frac{x^{2}}{- 1} + \frac{3 x}{- 1} + \frac{- 5}{- 1}$

Step 3.2.2

Divide $| 2 x |$ by $1$ .

$| 2 x | = \frac{x^{2}}{- 1} + \frac{3 x}{- 1} + \frac{- 5}{- 1}$

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 $\frac{x^{2}}{- 1}$ .

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

Step 3.3.1.2

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

$| 2 x | = - x^{2} + \frac{3 x}{- 1} + \frac{- 5}{- 1}$

Step 3.3.1.3

Move the negative one from the denominator of $\frac{3 x}{- 1}$ .

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

Step 3.3.1.4

Rewrite $- 1 \cdot (3 x)$ as $- (3 x)$ .

$| 2 x | = - x^{2} - (3 x) + \frac{- 5}{- 1}$

Step 3.3.1.5

Multiply $3$ by $- 1$ .

$| 2 x | = - x^{2} - 3 x + \frac{- 5}{- 1}$

Step 3.3.1.6

Divide $- 5$ by $- 1$ .

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

Step 4

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

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

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.

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

Step 5.2

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

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

Step 5.3

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

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Step 5.3.1

Subtract $2 x$ from both sides of the equation.

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

Step 5.3.2

Subtract $2 x$ from $- 3 x$ .

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

Step 5.4

Use the quadratic formula to find the solutions.

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

Step 5.5

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

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

Step 5.6

Simplify.

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Step 5.6.1

Simplify the numerator.

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Step 5.6.1.1

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

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

Step 5.6.1.2

Multiply $- 4 \cdot - 1 \cdot 5$ .

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Step 5.6.1.2.1

Multiply $- 4$ by $- 1$ .

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

Step 5.6.1.2.2

Multiply $4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 + 20}}{2 \cdot - 1}$

Step 5.6.1.3

Add $25$ and $20$ .

$x = \frac{5 \pm \sqrt{45}}{2 \cdot - 1}$

Step 5.6.1.4

Rewrite $45$ as $3^{2} \cdot 5$ .

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Step 5.6.1.4.1

Factor $9$ out of $45$ .

$x = \frac{5 \pm \sqrt{9 (5)}}{2 \cdot - 1}$

Step 5.6.1.4.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{5 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot - 1}$

Step 5.6.1.5

Pull terms out from under the radical.

$x = \frac{5 \pm 3 \sqrt{5}}{2 \cdot - 1}$

Step 5.6.2

Multiply $2$ by $- 1$ .

$x = \frac{5 \pm 3 \sqrt{5}}{- 2}$

Step 5.6.3

Move the negative in front of the fraction.

$x = - \frac{5 \pm 3 \sqrt{5}}{2}$

Step 5.7

The final answer is the combination of both solutions.

$x = - \frac{5 + 3 \sqrt{5}}{2}, - \frac{5 - 3 \sqrt{5}}{2}$

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

Simplify $- (- x^{2} - 3 x + 5)$ .

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Step 5.10.1

Rewrite.

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

Step 5.10.2

Simplify by adding zeros.

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

Step 5.10.3

Apply the distributive property.

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

Step 5.10.4

Simplify.

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Step 5.10.4.1

Multiply $- - x^{2}$ .

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Step 5.10.4.1.1

Multiply $- 1$ by $- 1$ .

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

Step 5.10.4.1.2

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

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

Step 5.10.4.2

Multiply $- 3$ by $- 1$ .

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

Step 5.10.4.3

Multiply $- 1$ by $5$ .

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

Step 5.11

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

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Step 5.11.1

Subtract $2 x$ from both sides of the equation.

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

Step 5.11.2

Subtract $2 x$ from $3 x$ .

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

Step 5.12

Use the quadratic formula to find the solutions.

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

Step 5.13

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

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

Step 5.14

Simplify.

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Step 5.14.1

Simplify the numerator.

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Step 5.14.1.1

One to any power is one.

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

Step 5.14.1.2

Multiply $- 4 \cdot 1 \cdot - 5$ .

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Step 5.14.1.2.1

Multiply $- 4$ by $1$ .

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

Step 5.14.1.2.2

Multiply $- 4$ by $- 5$ .

$x = \frac{- 1 \pm \sqrt{1 + 20}}{2 \cdot 1}$

Step 5.14.1.3

Add $1$ and $20$ .

$x = \frac{- 1 \pm \sqrt{21}}{2 \cdot 1}$

Step 5.14.2

Multiply $2$ by $1$ .

$x = \frac{- 1 \pm \sqrt{21}}{2}$

Step 5.15

The final answer is the combination of both solutions.

$x = - \frac{1 - \sqrt{21}}{2}, - \frac{1 + \sqrt{21}}{2}$

Step 5.16

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

$x = - \frac{5 + 3 \sqrt{5}}{2}, - \frac{5 - 3 \sqrt{5}}{2}, - \frac{1 - \sqrt{21}}{2}, - \frac{1 + \sqrt{21}}{2}$

Step 6

Exclude the solutions that do not make $x^{2} + 3 x + 7 = - | 2 x | + 12$ true.

$x = - \frac{5 - 3 \sqrt{5}}{2}, - \frac{1 + \sqrt{21}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5 - 3 \sqrt{5}}{2}, - \frac{1 + \sqrt{21}}{2}$

Decimal Form:

$x = 0.85410196 \dots, - 2.79128784 \dots$