Finite Math Examples

16 x^{2} + 25 = 40 x

$16 x^{2} + 25 = 40 x$

Step 1

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

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Step 1.1

Subtract

25

$25$ from both sides of the equation.

16 x^{2} = 40 x - 25

$16 x^{2} = 40 x - 25$

Step 1.2

Subtract

40 x

$40 x$ from both sides of the equation.

16 x^{2} - 40 x = - 25

$16 x^{2} - 40 x = - 25$

16 x^{2} - 40 x = - 25

$16 x^{2} - 40 x = - 25$

Step 2

Divide each term in

16 x^{2} - 40 x = - 25

$16 x^{2} - 40 x = - 25$ by

16

$16$ and simplify.

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Step 2.1

Divide each term in

16 x^{2} - 40 x = - 25

$16 x^{2} - 40 x = - 25$ by

16

$16$ .

\frac{16 x^{2}}{16} + \frac{- 40 x}{16} = \frac{- 25}{16}

$\frac{16 x^{2}}{16} + \frac{- 40 x}{16} = \frac{- 25}{16}$

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

16

$16$ .

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Step 2.2.1.1.1

Cancel the common factor.

\frac{16 x^{2}}{16} + \frac{- 40 x}{16} = \frac{- 25}{16}

$\frac{16 x^{2}}{16} + \frac{- 40 x}{16} = \frac{- 25}{16}$

Step 2.2.1.1.2

Divide

x^{2}

$x^{2}$ by

1

$1$ .

x^{2} + \frac{- 40 x}{16} = \frac{- 25}{16}

$x^{2} + \frac{- 40 x}{16} = \frac{- 25}{16}$

x^{2} + \frac{- 40 x}{16} = \frac{- 25}{16}

$x^{2} + \frac{- 40 x}{16} = \frac{- 25}{16}$

Step 2.2.1.2

Cancel the common factor of

- 40

$- 40$ and

16

$16$ .

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Step 2.2.1.2.1

Factor

8

$8$ out of

- 40 x

$- 40 x$ .

x^{2} + \frac{8 (- 5 x)}{16} = \frac{- 25}{16}

$x^{2} + \frac{8 (- 5 x)}{16} = \frac{- 25}{16}$

Step 2.2.1.2.2

Cancel the common factors.

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Step 2.2.1.2.2.1

Factor

8

$8$ out of

16

$16$ .

x^{2} + \frac{8 (- 5 x)}{8 (2)} = \frac{- 25}{16}

$x^{2} + \frac{8 (- 5 x)}{8 (2)} = \frac{- 25}{16}$

Step 2.2.1.2.2.2

Cancel the common factor.

x^{2} + \frac{8 (- 5 x)}{8 \cdot 2} = \frac{- 25}{16}

$x^{2} + \frac{8 (- 5 x)}{8 \cdot 2} = \frac{- 25}{16}$

Step 2.2.1.2.2.3

Rewrite the expression.

x^{2} + \frac{- 5 x}{2} = \frac{- 25}{16}

$x^{2} + \frac{- 5 x}{2} = \frac{- 25}{16}$

x^{2} + \frac{- 5 x}{2} = \frac{- 25}{16}

$x^{2} + \frac{- 5 x}{2} = \frac{- 25}{16}$

x^{2} + \frac{- 5 x}{2} = \frac{- 25}{16}

$x^{2} + \frac{- 5 x}{2} = \frac{- 25}{16}$

Step 2.2.1.3

Move the negative in front of the fraction.

x^{2} - \frac{5 x}{2} = \frac{- 25}{16}

$x^{2} - \frac{5 x}{2} = \frac{- 25}{16}$

x^{2} - \frac{5 x}{2} = \frac{- 25}{16}

$x^{2} - \frac{5 x}{2} = \frac{- 25}{16}$

x^{2} - \frac{5 x}{2} = \frac{- 25}{16}

$x^{2} - \frac{5 x}{2} = \frac{- 25}{16}$

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{5 x}{2} = - \frac{25}{16}

$x^{2} - \frac{5 x}{2} = - \frac{25}{16}$

x^{2} - \frac{5 x}{2} = - \frac{25}{16}

$x^{2} - \frac{5 x}{2} = - \frac{25}{16}$

x^{2} - \frac{5 x}{2} = - \frac{25}{16}

$x^{2} - \frac{5 x}{2} = - \frac{25}{16}$

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} = {(- \frac{5}{4})}^{2}

${(\frac{b}{2})}^{2} = {(- \frac{5}{4})}^{2}$

Step 4

Add the term to each side of the equation.

x^{2} - \frac{5 x}{2} + {(- \frac{5}{4})}^{2} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + {(- \frac{5}{4})}^{2} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

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}

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

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Step 5.1.1.1.1

Apply the product rule to

- \frac{5}{4}

$- \frac{5}{4}$ .

x^{2} - \frac{5 x}{2} + {(- 1)}^{2} {(\frac{5}{4})}^{2} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + {(- 1)}^{2} {(\frac{5}{4})}^{2} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

Step 5.1.1.1.2

Apply the product rule to

\frac{5}{4}

$\frac{5}{4}$ .

x^{2} - \frac{5 x}{2} + {(- 1)}^{2} \frac{5^{2}}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + {(- 1)}^{2} \frac{5^{2}}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

x^{2} - \frac{5 x}{2} + {(- 1)}^{2} \frac{5^{2}}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + {(- 1)}^{2} \frac{5^{2}}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

Step 5.1.1.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

x^{2} - \frac{5 x}{2} + 1 \frac{5^{2}}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + 1 \frac{5^{2}}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

Step 5.1.1.3

Multiply

\frac{5^{2}}{4^{2}}

$\frac{5^{2}}{4^{2}}$ by

1

$1$ .

x^{2} - \frac{5 x}{2} + \frac{5^{2}}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + \frac{5^{2}}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

Step 5.1.1.4

Raise

5

$5$ to the power of

2

$2$ .

x^{2} - \frac{5 x}{2} + \frac{25}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + \frac{25}{4^{2}} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

Step 5.1.1.5

Raise

4

$4$ to the power of

2

$2$ .

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- \frac{5}{4})}^{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify

- \frac{25}{16} + {(- \frac{5}{4})}^{2}

$- \frac{25}{16} + {(- \frac{5}{4})}^{2}$ .

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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}

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

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Step 5.2.1.1.1.1

Apply the product rule to

- \frac{5}{4}

$- \frac{5}{4}$ .

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- 1)}^{2} {(\frac{5}{4})}^{2}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- 1)}^{2} {(\frac{5}{4})}^{2}$

Step 5.2.1.1.1.2

Apply the product rule to

\frac{5}{4}

$\frac{5}{4}$ .

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- 1)}^{2} \frac{5^{2}}{4^{2}}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- 1)}^{2} \frac{5^{2}}{4^{2}}$

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- 1)}^{2} \frac{5^{2}}{4^{2}}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + {(- 1)}^{2} \frac{5^{2}}{4^{2}}$

Step 5.2.1.1.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + 1 \frac{5^{2}}{4^{2}}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + 1 \frac{5^{2}}{4^{2}}$

Step 5.2.1.1.3

Multiply

\frac{5^{2}}{4^{2}}

$\frac{5^{2}}{4^{2}}$ by

1

$1$ .

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + \frac{5^{2}}{4^{2}}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + \frac{5^{2}}{4^{2}}$

Step 5.2.1.1.4

Raise

5

$5$ to the power of

2

$2$ .

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + \frac{25}{4^{2}}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + \frac{25}{4^{2}}$

Step 5.2.1.1.5

Raise

4

$4$ to the power of

2

$2$ .

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + \frac{25}{16}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + \frac{25}{16}$

x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + \frac{25}{16}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = - \frac{25}{16} + \frac{25}{16}$

Step 5.2.1.2

Combine fractions.

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Step 5.2.1.2.1

Combine the numerators over the common denominator.

x^{2} - \frac{5 x}{2} + \frac{25}{16} = \frac{- 25 + 25}{16}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = \frac{- 25 + 25}{16}$

Step 5.2.1.2.2

Simplify the expression.

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Step 5.2.1.2.2.1

Add

- 25

$- 25$ and

25

$25$ .

x^{2} - \frac{5 x}{2} + \frac{25}{16} = \frac{0}{16}

$x^{2} - \frac{5 x}{2} + \frac{25}{16} = \frac{0}{16}$

Step 5.2.1.2.2.2

Divide

0

$0$ by

16

$16$ .

x^{2} - \frac{5 x}{2} + \frac{25}{16} = 0