Finite Math Examples

$x = - 2 (x^{2} - 10 x)$

Step 1

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

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Step 1.1

Simplify $- 2 (x^{2} - 10 x)$ .

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Step 1.1.1

Apply the distributive property.

$x = - 2 x^{2} - 2 (- 10 x)$

Step 1.1.2

Multiply $- 10$ by $- 2$ .

$x = - 2 x^{2} + 20 x$

Step 1.2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 2 x^{2} + 20 x = x$

Step 1.3

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

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Step 1.3.1

Subtract $x$ from both sides of the equation.

$- 2 x^{2} + 20 x - x = 0$

Step 1.3.2

Subtract $x$ from $20 x$ .

$- 2 x^{2} + 19 x = 0$

Step 2

Divide each term in $- 2 x^{2} + 19 x = 0$ by $- 2$ and simplify.

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Step 2.1

Divide each term in $- 2 x^{2} + 19 x = 0$ by $- 2$ .

$\frac{- 2 x^{2}}{- 2} + \frac{19 x}{- 2} = \frac{0}{- 2}$

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.

$\frac{- 2 x^{2}}{- 2} + \frac{19 x}{- 2} = \frac{0}{- 2}$

Step 2.2.1.1.2

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

$x^{2} + \frac{19 x}{- 2} = \frac{0}{- 2}$

Step 2.2.1.2

Move the negative in front of the fraction.

$x^{2} - \frac{19 x}{2} = \frac{0}{- 2}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Divide $0$ by $- 2$ .

$x^{2} - \frac{19 x}{2} = 0$

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 $b$ .

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

Step 4

Add the term to each side of the equation.

$x^{2} - \frac{19 x}{2} + {(- \frac{19}{4})}^{2} = 0 + {(- \frac{19}{4})}^{2}$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1.1.1

Apply the product rule to $- \frac{19}{4}$ .

$x^{2} - \frac{19 x}{2} + {(- 1)}^{2} {(\frac{19}{4})}^{2} = 0 + {(- \frac{19}{4})}^{2}$

Step 5.1.1.1.2

Apply the product rule to $\frac{19}{4}$ .

$x^{2} - \frac{19 x}{2} + {(- 1)}^{2} \frac{19^{2}}{4^{2}} = 0 + {(- \frac{19}{4})}^{2}$

Step 5.1.1.2

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

$x^{2} - \frac{19 x}{2} + 1 \frac{19^{2}}{4^{2}} = 0 + {(- \frac{19}{4})}^{2}$

Step 5.1.1.3

Multiply $\frac{19^{2}}{4^{2}}$ by $1$ .

$x^{2} - \frac{19 x}{2} + \frac{19^{2}}{4^{2}} = 0 + {(- \frac{19}{4})}^{2}$

Step 5.1.1.4

Raise $19$ to the power of $2$ .

$x^{2} - \frac{19 x}{2} + \frac{361}{4^{2}} = 0 + {(- \frac{19}{4})}^{2}$

Step 5.1.1.5

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

$x^{2} - \frac{19 x}{2} + \frac{361}{16} = 0 + {(- \frac{19}{4})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $0 + {(- \frac{19}{4})}^{2}$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1.1

Apply the product rule to $- \frac{19}{4}$ .

$x^{2} - \frac{19 x}{2} + \frac{361}{16} = 0 + {(- 1)}^{2} {(\frac{19}{4})}^{2}$

Step 5.2.1.1.1.2

Apply the product rule to $\frac{19}{4}$ .

$x^{2} - \frac{19 x}{2} + \frac{361}{16} = 0 + {(- 1)}^{2} \frac{19^{2}}{4^{2}}$

Step 5.2.1.1.2

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

$x^{2} - \frac{19 x}{2} + \frac{361}{16} = 0 + 1 \frac{19^{2}}{4^{2}}$

Step 5.2.1.1.3

Multiply $\frac{19^{2}}{4^{2}}$ by $1$ .

$x^{2} - \frac{19 x}{2} + \frac{361}{16} = 0 + \frac{19^{2}}{4^{2}}$

Step 5.2.1.1.4

Raise $19$ to the power of $2$ .

$x^{2} - \frac{19 x}{2} + \frac{361}{16} = 0 + \frac{361}{4^{2}}$

Step 5.2.1.1.5

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

$x^{2} - \frac{19 x}{2} + \frac{361}{16} = 0 + \frac{361}{16}$

Step 5.2.1.2

Add $0$ and $\frac{361}{16}$ .

$x^{2} - \frac{19 x}{2} + \frac{361}{16} = \frac{361}{16}$

Step 6

Factor the perfect trinomial square into ${(x - \frac{19}{4})}^{2}$ .

${(x - \frac{19}{4})}^{2} = \frac{361}{16}$

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.

$x - \frac{19}{4} = \pm \sqrt{\frac{361}{16}}$

Step 7.2

Simplify $\pm \sqrt{\frac{361}{16}}$ .

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Step 7.2.1

Rewrite $\sqrt{\frac{361}{16}}$ as $\frac{\sqrt{361}}{\sqrt{16}}$ .

$x - \frac{19}{4} = \pm \frac{\sqrt{361}}{\sqrt{16}}$

Step 7.2.2

Simplify the numerator.

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Step 7.2.2.1

Rewrite $361$ as $19^{2}$ .

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

Step 7.2.2.2

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

$x - \frac{19}{4} = \pm \frac{19}{\sqrt{16}}$

Step 7.2.3

Simplify the denominator.

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Step 7.2.3.1

Rewrite $16$ as $4^{2}$ .

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

Step 7.2.3.2

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

$x - \frac{19}{4} = \pm \frac{19}{4}$

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 $\pm$ to find the first solution.

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

Step 7.3.2

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

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Step 7.3.2.1

Add $\frac{19}{4}$ to both sides of the equation.

$x = \frac{19}{4} + \frac{19}{4}$

Step 7.3.2.2

Combine the numerators over the common denominator.

$x = \frac{19 + 19}{4}$

Step 7.3.2.3

Add $19$ and $19$ .

$x = \frac{38}{4}$

Step 7.3.2.4

Cancel the common factor of $38$ and $4$ .

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Step 7.3.2.4.1

Factor $2$ out of $38$ .

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

Step 7.3.2.4.2

Cancel the common factors.

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Step 7.3.2.4.2.1

Factor $2$ out of $4$ .

$x = \frac{2 \cdot 19}{2 \cdot 2}$

Step 7.3.2.4.2.2

Cancel the common factor.

$x = \frac{2 \cdot 19}{2 \cdot 2}$

Step 7.3.2.4.2.3

Rewrite the expression.

$x = \frac{19}{2}$

Step 7.3.3

Next, use the negative value of the $\pm$ to find the second solution.

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

Step 7.3.4

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

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Step 7.3.4.1

Add $\frac{19}{4}$ to both sides of the equation.

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

Step 7.3.4.2

Combine the numerators over the common denominator.

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

Step 7.3.4.3

Add $- 19$ and $19$ .

$x = \frac{0}{4}$

Step 7.3.4.4

Divide $0$ by $4$ .

$x = 0$

Step 7.3.5

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

$x = \frac{19}{2}, 0$