Algebra Examples

$2 | x - 1 | - x^{2} + 2 x + 7 = 0$

Step 1

Move all terms not containing $| x - 1 |$ to the right side of the equation.

Tap for more steps...

Step 1.1

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

$2 | x - 1 | + 2 x + 7 = x^{2}$

Step 1.2

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

$2 | x - 1 | + 7 = x^{2} - 2 x$

Step 1.3

Subtract $7$ from both sides of the equation.

$2 | x - 1 | = x^{2} - 2 x - 7$

Step 2

Divide each term in $2 | x - 1 | = x^{2} - 2 x - 7$ by $2$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $2 | x - 1 | = x^{2} - 2 x - 7$ by $2$ .

$\frac{2 | x - 1 |}{2} = \frac{x^{2}}{2} + \frac{- 2 x}{2} + \frac{- 7}{2}$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.1.1

Cancel the common factor.

$\frac{2 | x - 1 |}{2} = \frac{x^{2}}{2} + \frac{- 2 x}{2} + \frac{- 7}{2}$

Step 2.2.1.2

Divide $| x - 1 |$ by $1$ .

$| x - 1 | = \frac{x^{2}}{2} + \frac{- 2 x}{2} + \frac{- 7}{2}$

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Simplify each term.

Tap for more steps...

Step 2.3.1.1

Cancel the common factor of $- 2$ and $2$ .

Tap for more steps...

Step 2.3.1.1.1

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

$| x - 1 | = \frac{x^{2}}{2} + \frac{2 (- x)}{2} + \frac{- 7}{2}$

Step 2.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 2.3.1.1.2.1

Factor $2$ out of $2$ .

$| x - 1 | = \frac{x^{2}}{2} + \frac{2 (- x)}{2 (1)} + \frac{- 7}{2}$

Step 2.3.1.1.2.2

Cancel the common factor.

$| x - 1 | = \frac{x^{2}}{2} + \frac{2 (- x)}{2 \cdot 1} + \frac{- 7}{2}$

Step 2.3.1.1.2.3

Rewrite the expression.

$| x - 1 | = \frac{x^{2}}{2} + \frac{- x}{1} + \frac{- 7}{2}$

Step 2.3.1.1.2.4

Divide $- x$ by $1$ .

$| x - 1 | = \frac{x^{2}}{2} - x + \frac{- 7}{2}$

Step 2.3.1.2

Move the negative in front of the fraction.

$| x - 1 | = \frac{x^{2}}{2} - x - \frac{7}{2}$

Step 3

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

$x - 1 = \pm (\frac{x^{2}}{2} - x - \frac{7}{2})$

Step 4

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

Tap for more steps...

Step 4.1

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

$x - 1 = \frac{x^{2}}{2} - x - \frac{7}{2}$

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.

$\frac{x^{2}}{2} - x - \frac{7}{2} = x - 1$

Step 4.3

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

Tap for more steps...

Step 4.3.1

Subtract $x$ from both sides of the equation.

$\frac{x^{2}}{2} - x - \frac{7}{2} - x = - 1$

Step 4.3.2

Subtract $x$ from $- x$ .

$\frac{x^{2}}{2} - 2 x - \frac{7}{2} = - 1$

Step 4.4

Multiply each term in $\frac{x^{2}}{2} - 2 x - \frac{7}{2} = - 1$ by $2$ to eliminate the fractions.

Tap for more steps...

Step 4.4.1

Multiply each term in $\frac{x^{2}}{2} - 2 x - \frac{7}{2} = - 1$ by $2$ .

$\frac{x^{2}}{2} \cdot 2 - 2 x \cdot 2 - \frac{7}{2} \cdot 2 = - 1 \cdot 2$

Step 4.4.2

Simplify the left side.

Tap for more steps...

Step 4.4.2.1

Simplify each term.

Tap for more steps...

Step 4.4.2.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.2.1.1.1

Cancel the common factor.

$\frac{x^{2}}{2} \cdot 2 - 2 x \cdot 2 - \frac{7}{2} \cdot 2 = - 1 \cdot 2$

Step 4.4.2.1.1.2

Rewrite the expression.

$x^{2} - 2 x \cdot 2 - \frac{7}{2} \cdot 2 = - 1 \cdot 2$

Step 4.4.2.1.2

Multiply $2$ by $- 2$ .

$x^{2} - 4 x - \frac{7}{2} \cdot 2 = - 1 \cdot 2$

Step 4.4.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.2.1.3.1

Move the leading negative in $- \frac{7}{2}$ into the numerator.

$x^{2} - 4 x + \frac{- 7}{2} \cdot 2 = - 1 \cdot 2$

Step 4.4.2.1.3.2

Cancel the common factor.

$x^{2} - 4 x + \frac{- 7}{2} \cdot 2 = - 1 \cdot 2$

Step 4.4.2.1.3.3

Rewrite the expression.

$x^{2} - 4 x - 7 = - 1 \cdot 2$

Step 4.4.3

Simplify the right side.

Tap for more steps...

Step 4.4.3.1

Multiply $- 1$ by $2$ .

$x^{2} - 4 x - 7 = - 2$

Step 4.5

Add $2$ to both sides of the equation.

$x^{2} - 4 x - 7 + 2 = 0$

Step 4.6

Add $- 7$ and $2$ .

$x^{2} - 4 x - 5 = 0$

Step 4.7

Factor $x^{2} - 4 x - 5$ using the AC method.

Tap for more steps...

Step 4.7.1

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 $- 5$ and whose sum is $- 4$ .

$- 5, 1$

Step 4.7.2

Write the factored form using these integers.

$(x - 5) (x + 1) = 0$

Step 4.8

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

$x - 5 = 0$

$x + 1 = 0$

Step 4.9

Set $x - 5$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 4.9.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 4.9.2

Add $5$ to both sides of the equation.

$x = 5$

Step 4.10

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

Tap for more steps...

Step 4.10.1

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

$x + 1 = 0$

Step 4.10.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 4.11

The final solution is all the values that make $(x - 5) (x + 1) = 0$ true.

$x = 5, - 1$

Step 4.12

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

$x - 1 = - (\frac{x^{2}}{2} - x - \frac{7}{2})$

Step 4.13

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

$- (\frac{x^{2}}{2} - x - \frac{7}{2}) = x - 1$

Step 4.14

Simplify $- (\frac{x^{2}}{2} - x - \frac{7}{2})$ .

Tap for more steps...

Step 4.14.1

Rewrite.

$0 + 0 - (\frac{x^{2}}{2} - x - \frac{7}{2}) = x - 1$

Step 4.14.2

Simplify by adding zeros.

$- (\frac{x^{2}}{2} - x - \frac{7}{2}) = x - 1$

Step 4.14.3

Apply the distributive property.

$- \frac{x^{2}}{2} - - x - - \frac{7}{2} = x - 1$

Step 4.14.4

Simplify.

Tap for more steps...

Step 4.14.4.1

Multiply $- - x$ .

Tap for more steps...

Step 4.14.4.1.1

Multiply $- 1$ by $- 1$ .

$- \frac{x^{2}}{2} + 1 x - - \frac{7}{2} = x - 1$

Step 4.14.4.1.2

Multiply $x$ by $1$ .

$- \frac{x^{2}}{2} + x - - \frac{7}{2} = x - 1$

Step 4.14.4.2

Multiply $- - \frac{7}{2}$ .

Tap for more steps...

Step 4.14.4.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{x^{2}}{2} + x + 1 (\frac{7}{2}) = x - 1$

Step 4.14.4.2.2

Multiply $\frac{7}{2}$ by $1$ .

$- \frac{x^{2}}{2} + x + \frac{7}{2} = x - 1$

Step 4.15

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

Tap for more steps...

Step 4.15.1

Subtract $x$ from both sides of the equation.

$- \frac{x^{2}}{2} + x + \frac{7}{2} - x = - 1$

Step 4.15.2

Combine the opposite terms in $- \frac{x^{2}}{2} + x + \frac{7}{2} - x$ .

Tap for more steps...

Step 4.15.2.1

Subtract $x$ from $x$ .

$- \frac{x^{2}}{2} + 0 + \frac{7}{2} = - 1$

Step 4.15.2.2

Add $- \frac{x^{2}}{2}$ and $0$ .

$- \frac{x^{2}}{2} + \frac{7}{2} = - 1$

Step 4.16

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

Tap for more steps...

Step 4.16.1

Subtract $\frac{7}{2}$ from both sides of the equation.

$- \frac{x^{2}}{2} = - 1 - \frac{7}{2}$

Step 4.16.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{x^{2}}{2} = - 1 \cdot \frac{2}{2} - \frac{7}{2}$

Step 4.16.3

Combine $- 1$ and $\frac{2}{2}$ .

$- \frac{x^{2}}{2} = \frac{- 1 \cdot 2}{2} - \frac{7}{2}$

Step 4.16.4

Combine the numerators over the common denominator.

$- \frac{x^{2}}{2} = \frac{- 1 \cdot 2 - 7}{2}$

Step 4.16.5

Simplify the numerator.

Tap for more steps...

Step 4.16.5.1

Multiply $- 1$ by $2$ .

$- \frac{x^{2}}{2} = \frac{- 2 - 7}{2}$

Step 4.16.5.2

Subtract $7$ from $- 2$ .

$- \frac{x^{2}}{2} = \frac{- 9}{2}$

Step 4.16.6

Move the negative in front of the fraction.

$- \frac{x^{2}}{2} = - \frac{9}{2}$

Step 4.17

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x^{2} = - 9$

Step 4.18

Divide each term in $- x^{2} = - 9$ by $- 1$ and simplify.

Tap for more steps...

Step 4.18.1

Divide each term in $- x^{2} = - 9$ by $- 1$ .

$\frac{- x^{2}}{- 1} = \frac{- 9}{- 1}$

Step 4.18.2

Simplify the left side.

Tap for more steps...

Step 4.18.2.1

Dividing two negative values results in a positive value.

$\frac{x^{2}}{1} = \frac{- 9}{- 1}$

Step 4.18.2.2

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

$x^{2} = \frac{- 9}{- 1}$

Step 4.18.3

Simplify the right side.

Tap for more steps...

Step 4.18.3.1

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

Step 4.19

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

$x = \pm \sqrt{9}$

Step 4.20

Simplify $\pm \sqrt{9}$ .

Tap for more steps...

Step 4.20.1

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

$x = \pm \sqrt{3^{2}}$

Step 4.20.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 3$

Step 4.21

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

Tap for more steps...

Step 4.21.1

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

$x = 3$

Step 4.21.2

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

$x = - 3$

Step 4.21.3

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

$x = 3, - 3$

Step 4.22

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

$x = 5, - 1, 3, - 3$

Step 5

Exclude the solutions that do not make $2 | x - 1 | - x^{2} + 2 x + 7 = 0$ true.

$x = 5, - 3$