Solve for x 2|x-1|-x^2+2x+7=0

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

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

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

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

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

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

Subtract $7$ from both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

Simplify the right side.

Simplify each term.

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

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $- x$ by $1$ .

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

Move the negative in front of the fraction.

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

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

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

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

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

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

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

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

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

Subtract $x$ from both sides of the equation.

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

Subtract $x$ from $- x$ .

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

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

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

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

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Multiply $2$ by $- 2$ .

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

Cancel the common factor of $2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Simplify the right side.

Multiply $- 1$ by $2$ .

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

Add $2$ to both sides of the equation.

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

Add $- 7$ and $2$ .

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

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

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 $- 4$ .

$- 5, 1$

Write the factored form using these integers.

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

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

$x - 5 = 0$

$x + 1 = 0$

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

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

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

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

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

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

$x = 5, - 1$

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

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

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

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

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

Rewrite.

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

Simplify by adding zeros.

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

Apply the distributive property.

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

Simplify.

Multiply $- - x$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $x$ by $1$ .

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

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

Subtract $x$ from both sides of the equation.

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

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

Subtract $x$ from $x$ .

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

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

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $- 1$ by $2$ .

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

Subtract $7$ from $- 2$ .

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

Move the negative in front of the fraction.

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

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

$- x^{2} = - 9$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

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

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

Simplify the right side.

Divide $- 9$ by $- 1$ .

$x^{2} = 9$

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

$x = \pm \sqrt{9}$

Simplify $\pm \sqrt{9}$ .

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

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

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

$x = \pm 3$

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

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

$x = 3$

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

$x = - 3$

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

$x = 3, - 3$

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

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

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

$x = 5, - 3$

Solve for x 2|x-1|-x^2+2x+7=0

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