Algebra Examples

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

Step 1

Simplify the equation into a proper form to complete the square.

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Step 1.1

Simplify $7 + 2 x - x^{2}$ .

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Step 1.1.1

Move $7$ .

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

Step 1.1.2

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

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

Step 1.2

Subtract $7$ from both sides of the equation.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Dividing two negative values results in a positive value.

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

Step 2.2.1.2

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

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

Step 2.2.1.3

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

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

Step 2.2.1.4

Rewrite $- 1 \cdot (2 x)$ as $- (2 x)$ .

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

Step 2.2.1.5

Multiply $2$ by $- 1$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 7$ by $- 1$ .

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

Step 3

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of $b$ .

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

Step 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Raise $- 1$ to the power of $2$ .

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $7 + {(- 1)}^{2}$ .

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Step 5.2.1.1

Raise $- 1$ to the power of $2$ .

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

Step 5.2.1.2

Add $7$ and $1$ .

$x^{2} - 2 x + 1 = 8$

Step 6

Factor the perfect trinomial square into ${(x - 1)}^{2}$ .

${(x - 1)}^{2} = 8$

Step 7

Solve the equation for $x$ .

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Step 7.1

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

$x - 1 = \pm \sqrt{8}$

Step 7.2

Simplify $\pm \sqrt{8}$ .

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Step 7.2.1

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 7.2.1.1

Factor $4$ out of $8$ .

$x - 1 = \pm \sqrt{4 (2)}$

Step 7.2.1.2

Rewrite $4$ as $2^{2}$ .

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

Step 7.2.2

Pull terms out from under the radical.

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

Step 7.3

Add $1$ to both sides of the equation.

$x = \pm 2 \sqrt{2} + 1$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \pm 2 \sqrt{2} + 1$

Decimal Form:

$x = 3.82842712 \dots, - 1.82842712 \dots$