Finite Math Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

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

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Move the negative in front of the fraction.

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

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 $b$ .

${(\frac{b}{2})}^{2} = {(\frac{1}{4})}^{2}$

Step 4

Add the term to each side of the equation.

$x^{2} + \frac{x}{2} + {(\frac{1}{4})}^{2} = - \frac{1}{2} + {(\frac{1}{4})}^{2}$

Step 5

Simplify the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Apply the product rule to $\frac{1}{4}$ .

$x^{2} + \frac{x}{2} + \frac{1^{2}}{4^{2}} = - \frac{1}{2} + {(\frac{1}{4})}^{2}$

Step 5.1.1.2

One to any power is one.

$x^{2} + \frac{x}{2} + \frac{1}{4^{2}} = - \frac{1}{2} + {(\frac{1}{4})}^{2}$

Step 5.1.1.3

Raise $4$ to the power of $2$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = - \frac{1}{2} + {(\frac{1}{4})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- \frac{1}{2} + {(\frac{1}{4})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Apply the product rule to $\frac{1}{4}$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = - \frac{1}{2} + \frac{1^{2}}{4^{2}}$

Step 5.2.1.1.2

One to any power is one.

$x^{2} + \frac{x}{2} + \frac{1}{16} = - \frac{1}{2} + \frac{1}{4^{2}}$

Step 5.2.1.1.3

Raise $4$ to the power of $2$ .

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

Step 5.2.1.2

To write $- \frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = - \frac{1}{2} \cdot \frac{8}{8} + \frac{1}{16}$

Step 5.2.1.3

Write each expression with a common denominator of $16$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.2.1.3.1

Multiply $\frac{1}{2}$ by $\frac{8}{8}$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = - \frac{8}{2 \cdot 8} + \frac{1}{16}$

Step 5.2.1.3.2

Multiply $2$ by $8$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = - \frac{8}{16} + \frac{1}{16}$

Step 5.2.1.4

Combine the numerators over the common denominator.

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{- 8 + 1}{16}$

Step 5.2.1.5

Add $- 8$ and $1$ .

$x^{2} + \frac{x}{2} + \frac{1}{16} = \frac{- 7}{16}$

Step 5.2.1.6

Move the negative in front of the fraction.

$x^{2} + \frac{x}{2} + \frac{1}{16} = - \frac{7}{16}$

Step 6

Factor the perfect trinomial square into ${(x + \frac{1}{4})}^{2}$ .

${(x + \frac{1}{4})}^{2} = - \frac{7}{16}$

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.

$x + \frac{1}{4} = \pm \sqrt{- \frac{7}{16}}$

Step 7.2

Simplify $\pm \sqrt{- \frac{7}{16}}$ .

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Step 7.2.1

Rewrite $- \frac{7}{16}$ as ${(\frac{i}{4})}^{2} \cdot 7$ .

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Step 7.2.1.1

Rewrite $- 1$ as $i^{2}$ .

$x + \frac{1}{4} = \pm \sqrt{i^{2} \frac{7}{16}}$

Step 7.2.1.2

Factor the perfect power $1^{2}$ out of $7$ .

$x + \frac{1}{4} = \pm \sqrt{i^{2} \frac{1^{2} \cdot 7}{16}}$

Step 7.2.1.3

Factor the perfect power $4^{2}$ out of $16$ .

$x + \frac{1}{4} = \pm \sqrt{i^{2} \frac{1^{2} \cdot 7}{4^{2} \cdot 1}}$

Step 7.2.1.4

Rearrange the fraction $\frac{1^{2} \cdot 7}{4^{2} \cdot 1}$ .

$x + \frac{1}{4} = \pm \sqrt{i^{2} ({(\frac{1}{4})}^{2} \cdot 7)}$

Step 7.2.1.5

Rewrite $i^{2} {(\frac{1}{4})}^{2}$ as ${(\frac{i}{4})}^{2}$ .

$x + \frac{1}{4} = \pm \sqrt{{(\frac{i}{4})}^{2} \cdot 7}$

Step 7.2.2

Pull terms out from under the radical.

$x + \frac{1}{4} = \pm \frac{i}{4} \sqrt{7}$

Step 7.2.3

Combine $\frac{i}{4}$ and $\sqrt{7}$ .

$x + \frac{1}{4} = \pm \frac{i \sqrt{7}}{4}$

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 $\frac{1}{4}$ from both sides of the equation.

$x = \pm \frac{i \sqrt{7}}{4} - \frac{1}{4}$

Step 7.3.2

Reorder $\pm \frac{i \sqrt{7}}{4}$ and $- \frac{1}{4}$ .

$x = - \frac{1}{4} \pm \frac{i \sqrt{7}}{4}$