Trigonometry Examples

4 x^{2} + 4 y^{2} - 16 x - 8 y - 5 = 0

$4 x^{2} + 4 y^{2} - 16 x - 8 y - 5 = 0$

Step 1

Add

5

$5$ to both sides of the equation.

4 x^{2} + 4 y^{2} - 16 x - 8 y = 5

$4 x^{2} + 4 y^{2} - 16 x - 8 y = 5$

Step 2

Divide both sides of the equation by

4

$4$ .

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

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

Step 3

Complete the square for

x^{2} - 4 x

$x^{2} - 4 x$ .

Tap for more steps...

Step 3.1

Use the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , to find the values of

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 1

$a = 1$

b = - 4

$b = - 4$

c = 0

$c = 0$

Step 3.2

Consider the vertex form of a parabola.

a {(x + d)}^{2} + e

$a {(x + d)}^{2} + e$

Step 3.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

Tap for more steps...

Step 3.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

d = \frac{- 4}{2 \cdot 1}

$d = \frac{- 4}{2 \cdot 1}$

Step 3.3.2

Cancel the common factor of

- 4

$- 4$ and

2

$2$ .

Tap for more steps...

Step 3.3.2.1

Factor

2

$2$ out of

- 4

$- 4$ .

d = \frac{2 \cdot - 2}{2 \cdot 1}

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 3.3.2.2

Cancel the common factors.

Tap for more steps...

Step 3.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

d = \frac{2 \cdot - 2}{2 (1)}

$d = \frac{2 \cdot - 2}{2 (1)}$

Step 3.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 3.3.2.2.3

Rewrite the expression.

$d = \frac{- 2}{1}$

Step 3.3.2.2.4

Divide

$- 2$ by

$1$ .

$d = - 2$

Step 3.4

Find the value of

$e$ using the formula

$e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps...

Step 3.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

$e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 4)}^{2}}{4 \cdot 1}$

Step 3.4.2

Simplify the right side.

Tap for more steps...

Step 3.4.2.1

Simplify each term.

Tap for more steps...

Step 3.4.2.1.1

Cancel the common factor of

${(- 4)}^{2}$ and

$4$ .

Tap for more steps...

Step 3.4.2.1.1.1

Rewrite

$- 4$ as

$- 1 (4)$ .

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 \cdot 1}$

Step 3.4.2.1.1.2

Apply the product rule to

$- 1 (4)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 1}$

Step 3.4.2.1.1.3

Raise

$- 1$ to the power of

$2$ .

$e = 0 - \frac{1 \cdot 4^{2}}{4 \cdot 1}$

Step 3.4.2.1.1.4

Multiply

$4^{2}$ by

$1$ .

$e = 0 - \frac{4^{2}}{4 \cdot 1}$

Step 3.4.2.1.1.5

Factor

$4$ out of

$4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 3.4.2.1.1.6

Cancel the common factors.

Tap for more steps...

Step 3.4.2.1.1.6.1

Factor

$4$ out of

$4 \cdot 1$ .

$e = 0 - \frac{4 \cdot 4}{4 (1)}$

Step 3.4.2.1.1.6.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 3.4.2.1.1.6.3

Rewrite the expression.

$e = 0 - \frac{4}{1}$

Step 3.4.2.1.1.6.4

Divide

$4$ by

$1$ .

$e = 0 - 1 \cdot 4$

Step 3.4.2.1.2

Multiply

$- 1$ by

$4$ .

$e = 0 - 4$

Step 3.4.2.2

Subtract

$4$ from

$0$ .

$e = - 4$

Step 3.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(x - 2)}^{2} - 4$ .

${(x - 2)}^{2} - 4$

Step 4

Substitute

${(x - 2)}^{2} - 4$ for

$x^{2} - 4 x$ in the equation

$x^{2} + y^{2} - 4 x - 2 y = \frac{5}{4}$ .

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

Step 5

Move

$- 4$ to the right side of the equation by adding

$4$ to both sides.

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

Step 6

Complete the square for

$y^{2} - 2 y$ .

Tap for more steps...

Step 6.1

Use the form

$a x^{2} + b x + c$ , to find the values of

$a$ ,

$b$ , and

$c$ .

$a = 1$

$b = - 2$

$c = 0$

Step 6.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 6.3

Find the value of

$d$ using the formula

$d = \frac{b}{2 a}$ .

Tap for more steps...

Step 6.3.1

Substitute the values of

$a$ and

$b$ into the formula

$d = \frac{b}{2 a}$ .

$d = \frac{- 2}{2 \cdot 1}$

Step 6.3.2

Cancel the common factor of

$- 2$ and

$2$ .

Tap for more steps...

Step 6.3.2.1

Factor

$2$ out of

$- 2$ .

$d = \frac{2 \cdot - 1}{2 \cdot 1}$

Step 6.3.2.2

Cancel the common factors.

Tap for more steps...

Step 6.3.2.2.1

Factor

$2$ out of

$2 \cdot 1$ .

$d = \frac{2 \cdot - 1}{2 (1)}$

Step 6.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 1}{2 \cdot 1}$

Step 6.3.2.2.3

Rewrite the expression.

$d = \frac{- 1}{1}$

Step 6.3.2.2.4

Divide

$- 1$ by

$1$ .

$d = - 1$

Step 6.4

Find the value of

$e$ using the formula

$e = c - \frac{b^{2}}{4 a}$ .

Tap for more steps...

Step 6.4.1

Substitute the values of

$c$ ,

$b$ and

$a$ into the formula

$e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 2)}^{2}}{4 \cdot 1}$

Step 6.4.2

Simplify the right side.

Tap for more steps...

Step 6.4.2.1

Simplify each term.

Tap for more steps...

Step 6.4.2.1.1

Raise

$- 2$ to the power of

$2$ .

$e = 0 - \frac{4}{4 \cdot 1}$

Step 6.4.2.1.2

Multiply

$4$ by

$1$ .

$e = 0 - \frac{4}{4}$

Step 6.4.2.1.3

Cancel the common factor of

$4$ .

Tap for more steps...

Step 6.4.2.1.3.1

Cancel the common factor.

$e = 0 - \frac{4}{4}$

Step 6.4.2.1.3.2

Rewrite the expression.

$e = 0 - 1 \cdot 1$

Step 6.4.2.1.4

Multiply

$- 1$ by

$1$ .

$e = 0 - 1$

Step 6.4.2.2

Subtract

$1$ from

$0$ .

$e = - 1$

Step 6.5

Substitute the values of

$a$ ,

$d$ , and

$e$ into the vertex form

${(y - 1)}^{2} - 1$ .

${(y - 1)}^{2} - 1$

Step 7

Substitute

${(y - 1)}^{2} - 1$ for

$y^{2} - 2 y$ in the equation

$x^{2} + y^{2} - 4 x - 2 y = \frac{5}{4}$ .

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

Step 8

Move

$- 1$ to the right side of the equation by adding

$1$ to both sides.

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

Step 9

Simplify

$\frac{5}{4} + 4 + 1$ .

Tap for more steps...

Step 9.1

Find the common denominator.

Tap for more steps...

Step 9.1.1

Write

$4$ as a fraction with denominator

$1$ .

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

Step 9.1.2

Multiply

$\frac{4}{1}$ by

$\frac{4}{4}$ .

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

Step 9.1.3

Multiply

$\frac{4}{1}$ by

$\frac{4}{4}$ .

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

Step 9.1.4

Write

$1$ as a fraction with denominator

$1$ .

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

Step 9.1.5

Multiply

$\frac{1}{1}$ by

$\frac{4}{4}$ .

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

Step 9.1.6

Multiply

$\frac{1}{1}$ by

$\frac{4}{4}$ .

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

Step 9.2

Combine the numerators over the common denominator.

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

Step 9.3

Simplify the expression.

Tap for more steps...

Step 9.3.1

Multiply

$4$ by

$4$ .

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

Step 9.3.2

Add

$5$ and

$16$ .

${(x - 2)}^{2} + {(y - 1)}^{2} = \frac{21 + 4}{4}$

Step 9.3.3

Add

$21$ and

$4$ .

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