Precalculus Examples

$7 x^{2} + x + 7 y^{2} + 7 y - 1 = 0$

Step 1

Add $1$ to both sides of the equation.

$7 x^{2} + x + 7 y^{2} + 7 y = 1$

Step 2

Divide both sides of the equation by $7$ .

$x^{2} + \frac{x}{7} + y^{2} + y = \frac{1}{7}$

Step 3

Complete the square for $x^{2} + \frac{x}{7}$ .

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Step 3.1

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

$a = 1$

$b = \frac{1}{7}$

$c = 0$

Step 3.2

Consider the vertex form of a parabola.

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

Step 3.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 3.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{\frac{1}{7}}{2 \cdot 1}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{1}{7} \cdot \frac{1}{2 \cdot 1}$

Step 3.3.2.2

Cancel the common factor of $1$ .

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Step 3.3.2.2.1

Cancel the common factor.

$d = \frac{1}{7} \cdot \frac{1}{2 \cdot 1}$

Step 3.3.2.2.2

Rewrite the expression.

$d = \frac{1}{7} \cdot \frac{1}{2}$

Step 3.3.2.3

Multiply $\frac{1}{7} \cdot \frac{1}{2}$ .

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Step 3.3.2.3.1

Multiply $\frac{1}{7}$ by $\frac{1}{2}$ .

$d = \frac{1}{7 \cdot 2}$

Step 3.3.2.3.2

Multiply $7$ by $2$ .

$d = \frac{1}{14}$

Step 3.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 3.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(\frac{1}{7})}^{2}}{4 \cdot 1}$

Step 3.4.2

Simplify the right side.

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Step 3.4.2.1

Simplify each term.

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Step 3.4.2.1.1

Simplify the numerator.

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Step 3.4.2.1.1.1

Apply the product rule to $\frac{1}{7}$ .

$e = 0 - \frac{\frac{1^{2}}{7^{2}}}{4 \cdot 1}$

Step 3.4.2.1.1.2

One to any power is one.

$e = 0 - \frac{\frac{1}{7^{2}}}{4 \cdot 1}$

Step 3.4.2.1.1.3

Raise $7$ to the power of $2$ .

$e = 0 - \frac{\frac{1}{49}}{4 \cdot 1}$

Step 3.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{\frac{1}{49}}{4}$

Step 3.4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (\frac{1}{49} \cdot \frac{1}{4})$

Step 3.4.2.1.4

Multiply $\frac{1}{49} \cdot \frac{1}{4}$ .

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Step 3.4.2.1.4.1

Multiply $\frac{1}{49}$ by $\frac{1}{4}$ .

$e = 0 - \frac{1}{49 \cdot 4}$

Step 3.4.2.1.4.2

Multiply $49$ by $4$ .

$e = 0 - \frac{1}{196}$

Step 3.4.2.2

Subtract $\frac{1}{196}$ from $0$ .

$e = - \frac{1}{196}$

Step 3.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x + \frac{1}{14})}^{2} - \frac{1}{196}$ .

${(x + \frac{1}{14})}^{2} - \frac{1}{196}$

Step 4

Substitute ${(x + \frac{1}{14})}^{2} - \frac{1}{196}$ for $x^{2} + \frac{x}{7}$ in the equation $x^{2} + \frac{x}{7} + y^{2} + y = \frac{1}{7}$ .

${(x + \frac{1}{14})}^{2} - \frac{1}{196} + y^{2} + y = \frac{1}{7}$

Step 5

Move $- \frac{1}{196}$ to the right side of the equation by adding $\frac{1}{196}$ to both sides.

${(x + \frac{1}{14})}^{2} + y^{2} + y = \frac{1}{7} + \frac{1}{196}$

Step 6

Complete the square for $y^{2} + y$ .

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Step 6.1

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

$a = 1$

$b = 1$

$c = 0$

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

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Step 6.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

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

Step 6.3.2

Cancel the common factor of $1$ .

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Step 6.3.2.1

Cancel the common factor.

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

Step 6.3.2.2

Rewrite the expression.

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

Step 6.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 6.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

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

Step 6.4.2

Simplify the right side.

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Step 6.4.2.1

Simplify each term.

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Step 6.4.2.1.1

One to any power is one.

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

Step 6.4.2.1.2

Multiply $4$ by $1$ .

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

Step 6.4.2.2

Subtract $\frac{1}{4}$ from $0$ .

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

Step 6.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y + \frac{1}{2})}^{2} - \frac{1}{4}$ .

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

Step 7

Substitute ${(y + \frac{1}{2})}^{2} - \frac{1}{4}$ for $y^{2} + y$ in the equation $x^{2} + \frac{x}{7} + y^{2} + y = \frac{1}{7}$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} - \frac{1}{4} = \frac{1}{7} + \frac{1}{196}$

Step 8

Move $- \frac{1}{4}$ to the right side of the equation by adding $\frac{1}{4}$ to both sides.

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{1}{7} + \frac{1}{196} + \frac{1}{4}$

Step 9

Simplify $\frac{1}{7} + \frac{1}{196} + \frac{1}{4}$ .

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Step 9.1

Find the common denominator.

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Step 9.1.1

Multiply $\frac{1}{7}$ by $\frac{28}{28}$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{1}{7} \cdot \frac{28}{28} + \frac{1}{196} + \frac{1}{4}$

Step 9.1.2

Multiply $\frac{1}{7}$ by $\frac{28}{28}$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{28}{7 \cdot 28} + \frac{1}{196} + \frac{1}{4}$

Step 9.1.3

Multiply $\frac{1}{4}$ by $\frac{49}{49}$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{28}{7 \cdot 28} + \frac{1}{196} + \frac{1}{4} \cdot \frac{49}{49}$

Step 9.1.4

Multiply $\frac{1}{4}$ by $\frac{49}{49}$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{28}{7 \cdot 28} + \frac{1}{196} + \frac{49}{4 \cdot 49}$

Step 9.1.5

Multiply $7$ by $28$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{28}{196} + \frac{1}{196} + \frac{49}{4 \cdot 49}$

Step 9.1.6

Multiply $4$ by $49$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{28}{196} + \frac{1}{196} + \frac{49}{196}$

Step 9.2

Combine the numerators over the common denominator.

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{28 + 1 + 49}{196}$

Step 9.3

Simplify by adding numbers.

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Step 9.3.1

Add $28$ and $1$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{29 + 49}{196}$

Step 9.3.2

Add $29$ and $49$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{78}{196}$

Step 9.4

Cancel the common factor of $78$ and $196$ .

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Step 9.4.1

Factor $2$ out of $78$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{2 (39)}{196}$

Step 9.4.2

Cancel the common factors.

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Step 9.4.2.1

Factor $2$ out of $196$ .

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{2 \cdot 39}{2 \cdot 98}$

Step 9.4.2.2

Cancel the common factor.

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{2 \cdot 39}{2 \cdot 98}$

Step 9.4.2.3

Rewrite the expression.

${(x + \frac{1}{14})}^{2} + {(y + \frac{1}{2})}^{2} = \frac{39}{98}$

Step 10

This is the form of a circle. Use this form to determine the center and radius of the circle.

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$

Step 11

Match the values in this circle to those of the standard form. The variable $r$ represents the radius of the circle, $h$ represents the x-offset from the origin, and $k$ represents the y-offset from origin.

$r = \frac{\sqrt{78}}{14}$

$h = - \frac{1}{14}$

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

Step 12

The center of the circle is found at $(h, k)$ .

Center: $(- \frac{1}{14}, - \frac{1}{2})$

Step 13

These values represent the important values for graphing and analyzing a circle.

Center: $(- \frac{1}{14}, - \frac{1}{2})$

Radius: $\frac{\sqrt{78}}{14}$

Step 14