Algebra Examples

$2 x^{2} + 42 x + 260 = 6 x$

Step 1

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

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Step 1.1

Subtract $260$ from both sides of the equation.

$2 x^{2} + 42 x = 6 x - 260$

Step 1.2

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

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Step 1.2.1

Subtract $6 x$ from both sides of the equation.

$2 x^{2} + 42 x - 6 x = - 260$

Step 1.2.2

Subtract $6 x$ from $42 x$ .

$2 x^{2} + 36 x = - 260$

Step 2

Divide each term in $2 x^{2} + 36 x = - 260$ by $2$ and simplify.

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Step 2.1

Divide each term in $2 x^{2} + 36 x = - 260$ by $2$ .

$\frac{2 x^{2}}{2} + \frac{36 x}{2} = \frac{- 260}{2}$

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

Cancel the common factor of $2$ .

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Step 2.2.1.1.1

Cancel the common factor.

$\frac{2 x^{2}}{2} + \frac{36 x}{2} = \frac{- 260}{2}$

Step 2.2.1.1.2

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

$x^{2} + \frac{36 x}{2} = \frac{- 260}{2}$

Step 2.2.1.2

Cancel the common factor of $36$ and $2$ .

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Step 2.2.1.2.1

Factor $2$ out of $36 x$ .

$x^{2} + \frac{2 (18 x)}{2} = \frac{- 260}{2}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor $2$ out of $2$ .

$x^{2} + \frac{2 (18 x)}{2 (1)} = \frac{- 260}{2}$

Step 2.2.1.2.2.2

Cancel the common factor.

$x^{2} + \frac{2 (18 x)}{2 \cdot 1} = \frac{- 260}{2}$

Step 2.2.1.2.2.3

Rewrite the expression.

$x^{2} + \frac{18 x}{1} = \frac{- 260}{2}$

Step 2.2.1.2.2.4

Divide $18 x$ by $1$ .

$x^{2} + 18 x = \frac{- 260}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $- 260$ by $2$ .

$x^{2} + 18 x = - 130$

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 $b$ .

${(\frac{b}{2})}^{2} = {(9)}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + 18 x + {(9)}^{2} = - 130 + {(9)}^{2}$

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 $9$ to the power of $2$ .

$x^{2} + 18 x + 81 = - 130 + {(9)}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 130 + {(9)}^{2}$ .

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Step 5.2.1.1

Raise $9$ to the power of $2$ .

$x^{2} + 18 x + 81 = - 130 + 81$

Step 5.2.1.2

Add $- 130$ and $81$ .

$x^{2} + 18 x + 81 = - 49$

Step 6

Factor the perfect trinomial square into ${(x + 9)}^{2}$ .

${(x + 9)}^{2} = - 49$

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.

$x + 9 = \pm \sqrt{- 49}$

Step 7.2

Simplify $\pm \sqrt{- 49}$ .

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Step 7.2.1

Rewrite $- 49$ as $- 1 (49)$ .

$x + 9 = \pm \sqrt{- 1 (49)}$

Step 7.2.2

Rewrite $\sqrt{- 1 (49)}$ as $\sqrt{- 1} \cdot \sqrt{49}$ .

$x + 9 = \pm \sqrt{- 1} \cdot \sqrt{49}$

Step 7.2.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x + 9 = \pm i \cdot \sqrt{49}$

Step 7.2.4

Rewrite $49$ as $7^{2}$ .

$x + 9 = \pm i \cdot \sqrt{7^{2}}$

Step 7.2.5

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

$x + 9 = \pm i \cdot 7$

Step 7.2.6

Move $7$ to the left of $i$ .

$x + 9 = \pm 7 i$

Step 7.3

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

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Step 7.3.1

Subtract $9$ from both sides of the equation.

$x = \pm 7 i - 9$

Step 7.3.2

Reorder $\pm 7 i$ and $- 9$ .

$x = - 9 \pm 7 i$