Algebra Examples

$- 4 x^{2} + 16 x = 19$

Step 1

Divide each term in $- 4 x^{2} + 16 x = 19$ by $- 4$ and simplify.

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Step 1.1

Divide each term in $- 4 x^{2} + 16 x = 19$ by $- 4$ .

$\frac{- 4 x^{2}}{- 4} + \frac{16 x}{- 4} = \frac{19}{- 4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Cancel the common factor of $- 4$ .

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Step 1.2.1.1.1

Cancel the common factor.

$\frac{- 4 x^{2}}{- 4} + \frac{16 x}{- 4} = \frac{19}{- 4}$

Step 1.2.1.1.2

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

$x^{2} + \frac{16 x}{- 4} = \frac{19}{- 4}$

Step 1.2.1.2

Cancel the common factor of $16$ and $- 4$ .

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Step 1.2.1.2.1

Factor $4$ out of $16 x$ .

$x^{2} + \frac{4 (4 x)}{- 4} = \frac{19}{- 4}$

Step 1.2.1.2.2

Move the negative one from the denominator of $\frac{4 x}{- 1}$ .

$x^{2} - 1 \cdot (4 x) = \frac{19}{- 4}$

Step 1.2.1.3

Rewrite $- 1 \cdot (4 x)$ as $- (4 x)$ .

$x^{2} - (4 x) = \frac{19}{- 4}$

Step 1.2.1.4

Multiply $4$ by $- 1$ .

$x^{2} - 4 x = \frac{19}{- 4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative in front of the fraction.

$x^{2} - 4 x = - \frac{19}{4}$

Step 2

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

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

Step 3

Add the term to each side of the equation.

$x^{2} - 4 x + {(- 2)}^{2} = - \frac{19}{4} + {(- 2)}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

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

$x^{2} - 4 x + 4 = - \frac{19}{4} + {(- 2)}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{19}{4} + {(- 2)}^{2}$ .

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Step 4.2.1.1

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

$x^{2} - 4 x + 4 = - \frac{19}{4} + 4$

Step 4.2.1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} - 4 x + 4 = - \frac{19}{4} + 4 \cdot \frac{4}{4}$

Step 4.2.1.3

Combine $4$ and $\frac{4}{4}$ .

$x^{2} - 4 x + 4 = - \frac{19}{4} + \frac{4 \cdot 4}{4}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x^{2} - 4 x + 4 = \frac{- 19 + 4 \cdot 4}{4}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $4$ by $4$ .

$x^{2} - 4 x + 4 = \frac{- 19 + 16}{4}$

Step 4.2.1.5.2

Add $- 19$ and $16$ .

$x^{2} - 4 x + 4 = \frac{- 3}{4}$

Step 4.2.1.6

Move the negative in front of the fraction.

$x^{2} - 4 x + 4 = - \frac{3}{4}$

Step 5

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

${(x - 2)}^{2} = - \frac{3}{4}$

Step 6

Solve the equation for $x$ .

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Step 6.1

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

$x - 2 = \pm \sqrt{- \frac{3}{4}}$

Step 6.2

Simplify $\pm \sqrt{- \frac{3}{4}}$ .

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Step 6.2.1

Rewrite $- \frac{3}{4}$ as ${(\frac{i}{2})}^{2} \cdot 3$ .

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Step 6.2.1.1

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

$x - 2 = \pm \sqrt{i^{2} \frac{3}{4}}$

Step 6.2.1.2

Factor the perfect power $1^{2}$ out of $3$ .

$x - 2 = \pm \sqrt{i^{2} \frac{1^{2} \cdot 3}{4}}$

Step 6.2.1.3

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

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

Step 6.2.1.4

Rearrange the fraction $\frac{1^{2} \cdot 3}{2^{2} \cdot 1}$ .

$x - 2 = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \cdot 3)}$

Step 6.2.1.5

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

$x - 2 = \pm \sqrt{{(\frac{i}{2})}^{2} \cdot 3}$

Step 6.2.2

Pull terms out from under the radical.

$x - 2 = \pm \frac{i}{2} \sqrt{3}$

Step 6.2.3

Combine $\frac{i}{2}$ and $\sqrt{3}$ .

$x - 2 = \pm \frac{i \sqrt{3}}{2}$

Step 6.3

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

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Step 6.3.1

Add $2$ to both sides of the equation.

$x = \pm \frac{i \sqrt{3}}{2} + 2$

Step 6.3.2

Reorder $\pm \frac{i \sqrt{3}}{2}$ and $2$ .

$x = 2 \pm \frac{i \sqrt{3}}{2}$