Finite Math Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Subtract $25$ from both sides of the equation.

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

Step 1.2

Subtract $40 x$ from both sides of the equation.

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

Step 2

Divide each term in $16 x^{2} - 40 x = - 25$ by $16$ and simplify.

Tap for more steps...

Step 2.1

Divide each term in $16 x^{2} - 40 x = - 25$ by $16$ .

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

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1

Cancel the common factor of $16$ .

Tap for more steps...

Step 2.2.1.1.1

Cancel the common factor.

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

Step 2.2.1.1.2

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

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

Step 2.2.1.2

Cancel the common factor of $- 40$ and $16$ .

Tap for more steps...

Step 2.2.1.2.1

Factor $8$ out of $- 40 x$ .

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

Step 2.2.1.2.2

Cancel the common factors.

Tap for more steps...

Step 2.2.1.2.2.1

Factor $8$ out of $16$ .

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

Step 2.2.1.2.2.2

Cancel the common factor.

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

Step 2.2.1.2.2.3

Rewrite the expression.

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

Step 2.2.1.3

Move the negative in front of the fraction.

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

Step 2.3

Simplify the right side.

Tap for more steps...

Step 2.3.1

Move the negative in front of the fraction.

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

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 $b$ .

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

Step 4

Add the term to each side of the equation.

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

Step 5

Simplify the equation.

Tap for more steps...

Step 5.1

Simplify the left side.

Tap for more steps...

Step 5.1.1

Simplify each term.

Tap for more steps...

Step 5.1.1.1

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

Tap for more steps...

Step 5.1.1.1.1

Apply the product rule to $- \frac{5}{4}$ .

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

Step 5.1.1.1.2

Apply the product rule to $\frac{5}{4}$ .

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

Step 5.1.1.2

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

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

Step 5.1.1.3

Multiply $\frac{5^{2}}{4^{2}}$ by $1$ .

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

Step 5.1.1.4

Raise $5$ to the power of $2$ .

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

Step 5.1.1.5

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

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

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

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

Tap for more steps...

Step 5.2.1.1

Simplify each term.

Tap for more steps...

Step 5.2.1.1.1

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

Tap for more steps...

Step 5.2.1.1.1.1

Apply the product rule to $- \frac{5}{4}$ .

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

Step 5.2.1.1.1.2

Apply the product rule to $\frac{5}{4}$ .

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

Step 5.2.1.1.2

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

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

Step 5.2.1.1.3

Multiply $\frac{5^{2}}{4^{2}}$ by $1$ .

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

Step 5.2.1.1.4

Raise $5$ to the power of $2$ .

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

Step 5.2.1.1.5

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

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

Step 5.2.1.2

Combine fractions.

Tap for more steps...

Step 5.2.1.2.1

Combine the numerators over the common denominator.

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

Step 5.2.1.2.2

Simplify the expression.

Tap for more steps...

Step 5.2.1.2.2.1

Add $- 25$ and $25$ .

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

Step 5.2.1.2.2.2

Divide $0$ by $16$ .

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

Step 6

Factor the perfect trinomial square into ${(x - \frac{5}{4})}^{2}$ .

${(x - \frac{5}{4})}^{2} = 0$

Step 7

Solve the equation for $x$ .

Tap for more steps...

Step 7.1

Set the $x - \frac{5}{4}$ equal to $0$ .

$x - \frac{5}{4} = 0$

Step 7.2

Add $\frac{5}{4}$ to both sides of the equation.

$x = \frac{5}{4}$