Algebra Examples

2 x^{2} - 4 x - 16 = 0

$2 x^{2} - 4 x - 16 = 0$

Step 1

Add

16

$16$ to both sides of the equation.

2 x^{2} - 4 x = 16

$2 x^{2} - 4 x = 16$

Step 2

Divide each term in

2 x^{2} - 4 x = 16

$2 x^{2} - 4 x = 16$ by

2

$2$ and simplify.

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Step 2.1

Divide each term in

2 x^{2} - 4 x = 16

$2 x^{2} - 4 x = 16$ by

2

$2$ .

\frac{2 x^{2}}{2} + \frac{- 4 x}{2} = \frac{16}{2}

$\frac{2 x^{2}}{2} + \frac{- 4 x}{2} = \frac{16}{2}$

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

$2$ .

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Step 2.2.1.1.1

Cancel the common factor.

\frac{2 x^{2}}{2} + \frac{- 4 x}{2} = \frac{16}{2}

$\frac{2 x^{2}}{2} + \frac{- 4 x}{2} = \frac{16}{2}$

Step 2.2.1.1.2

Divide

x^{2}

$x^{2}$ by

1

$1$ .

x^{2} + \frac{- 4 x}{2} = \frac{16}{2}

$x^{2} + \frac{- 4 x}{2} = \frac{16}{2}$

x^{2} + \frac{- 4 x}{2} = \frac{16}{2}

$x^{2} + \frac{- 4 x}{2} = \frac{16}{2}$

Step 2.2.1.2

Cancel the common factor of

- 4

$- 4$ and

2

$2$ .

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Step 2.2.1.2.1

Factor

2

$2$ out of

- 4 x

$- 4 x$ .

x^{2} + \frac{2 (- 2 x)}{2} = \frac{16}{2}

$x^{2} + \frac{2 (- 2 x)}{2} = \frac{16}{2}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor

2

$2$ out of

2

$2$ .

x^{2} + \frac{2 (- 2 x)}{2 (1)} = \frac{16}{2}

$x^{2} + \frac{2 (- 2 x)}{2 (1)} = \frac{16}{2}$

Step 2.2.1.2.2.2

Cancel the common factor.

x^{2} + \frac{2 (- 2 x)}{2 \cdot 1} = \frac{16}{2}

$x^{2} + \frac{2 (- 2 x)}{2 \cdot 1} = \frac{16}{2}$

Step 2.2.1.2.2.3

Rewrite the expression.

x^{2} + \frac{- 2 x}{1} = \frac{16}{2}

$x^{2} + \frac{- 2 x}{1} = \frac{16}{2}$

Step 2.2.1.2.2.4

Divide

- 2 x

$- 2 x$ by

1

$1$ .

x^{2} - 2 x = \frac{16}{2}

$x^{2} - 2 x = \frac{16}{2}$

x^{2} - 2 x = \frac{16}{2}

$x^{2} - 2 x = \frac{16}{2}$

x^{2} - 2 x = \frac{16}{2}

$x^{2} - 2 x = \frac{16}{2}$

x^{2} - 2 x = \frac{16}{2}

$x^{2} - 2 x = \frac{16}{2}$

x^{2} - 2 x = \frac{16}{2}

$x^{2} - 2 x = \frac{16}{2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide

16

$16$ by

2

$2$ .

x^{2} - 2 x = 8

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

x^{2} - 2 x = 8

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

x^{2} - 2 x = 8

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

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

$b$ .

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

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

Step 4

Add the term to each side of the equation.

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

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

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

$- 1$ to the power of

2

$2$ .

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

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

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

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify

8 + {(- 1)}^{2}

$8 + {(- 1)}^{2}$ .

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Step 5.2.1.1

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

Step 5.2.1.2

Add

8

$8$ and

1

$1$ .

x^{2} - 2 x + 1 = 9

$x^{2} - 2 x + 1 = 9$

x^{2} - 2 x + 1 = 9

$x^{2} - 2 x + 1 = 9$

x^{2} - 2 x + 1 = 9

$x^{2} - 2 x + 1 = 9$

x^{2} - 2 x + 1 = 9

$x^{2} - 2 x + 1 = 9$

Step 6

Factor the perfect trinomial square into

{(x - 1)}^{2}

${(x - 1)}^{2}$ .

{(x - 1)}^{2} = 9

${(x - 1)}^{2} = 9$

Step 7

Solve the equation for

x

$x$ .

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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{9}

$x - 1 = \pm \sqrt{9}$

Step 7.2

Simplify

\pm \sqrt{9}

$\pm \sqrt{9}$ .

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Step 7.2.1

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

x - 1 = \pm \sqrt{3^{2}}

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

Step 7.2.2

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

x - 1 = \pm 3

$x - 1 = \pm 3$

x - 1 = \pm 3

$x - 1 = \pm 3$

Step 7.3

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

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Step 7.3.1

First, use the positive value of the

\pm

$\pm$ to find the first solution.

x - 1 = 3

$x - 1 = 3$

Step 7.3.2

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 7.3.2.1

Add

1

$1$ to both sides of the equation.

x = 3 + 1

$x = 3 + 1$

Step 7.3.2.2

Add

3

$3$ and

1

$1$ .

x = 4

$x = 4$

x = 4

$x = 4$

Step 7.3.3

Next, use the negative value of the

\pm

$\pm$ to find the second solution.

x - 1 = - 3

$x - 1 = - 3$

Step 7.3.4

Move all terms not containing

x

$x$ to the right side of the equation.

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Step 7.3.4.1

Add

1

$1$ to both sides of the equation.

x = - 3 + 1

$x = - 3 + 1$

Step 7.3.4.2

Add

- 3

$- 3$ and

1

$1$ .

x = - 2

$x = - 2$

x = - 2

$x = - 2$

Step 7.3.5

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

x = 4, - 2

$x = 4, - 2$

x = 4, - 2

$x = 4, - 2$

x = 4, - 2

$x = 4, - 2$

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