Precalculus Examples

${(x - 1)}^{2} + {(y - 3)}^{2} = x$

Step 1

Subtract $x$ from both sides of the equation.

${(x - 1)}^{2} + {(y - 3)}^{2} - x = 0$

Step 2

Complete the square for ${(x - 1)}^{2} - x$ .

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Step 2.1

Simplify the expression.

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$(x - 1) (x - 1) - x$

Step 2.1.1.2

Expand $(x - 1) (x - 1)$ using the FOIL Method.

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Step 2.1.1.2.1

Apply the distributive property.

$x (x - 1) - 1 (x - 1) - x$

Step 2.1.1.2.2

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 (x - 1) - x$

Step 2.1.1.2.3

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - x$

Step 2.1.1.3

Simplify and combine like terms.

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Step 2.1.1.3.1

Simplify each term.

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Step 2.1.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - x$

Step 2.1.1.3.1.2

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - x$

Step 2.1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - 1 x - 1 \cdot - 1 - x$

Step 2.1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - x - 1 \cdot - 1 - x$

Step 2.1.1.3.1.5

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 - x$

Step 2.1.1.3.2

Subtract $x$ from $- x$ .

$x^{2} - 2 x + 1 - x$

Step 2.1.2

Subtract $x$ from $- 2 x$ .

$x^{2} - 3 x + 1$

Step 2.2

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

$a = 1$

$b = - 3$

$c = 1$

Step 2.3

Consider the vertex form of a parabola.

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

Step 2.4

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

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Step 2.4.1

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

$d = \frac{- 3}{2 \cdot 1}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Multiply $2$ by $1$ .

$d = \frac{- 3}{2}$

Step 2.4.2.2

Move the negative in front of the fraction.

$d = - \frac{3}{2}$

Step 2.5

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

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Step 2.5.1

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

$e = 1 - \frac{{(- 3)}^{2}}{4 \cdot 1}$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Raise $- 3$ to the power of $2$ .

$e = 1 - \frac{9}{4 \cdot 1}$

Step 2.5.2.1.2

Multiply $4$ by $1$ .

$e = 1 - \frac{9}{4}$

Step 2.5.2.2

Write $1$ as a fraction with a common denominator.

$e = \frac{4}{4} - \frac{9}{4}$

Step 2.5.2.3

Combine the numerators over the common denominator.

$e = \frac{4 - 9}{4}$

Step 2.5.2.4

Subtract $9$ from $4$ .

$e = \frac{- 5}{4}$

Step 2.5.2.5

Move the negative in front of the fraction.

$e = - \frac{5}{4}$

Step 2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{3}{2})}^{2} - \frac{5}{4}$ .

${(x - \frac{3}{2})}^{2} - \frac{5}{4}$

Step 3

Substitute ${(x - \frac{3}{2})}^{2} - \frac{5}{4}$ for ${(x - 1)}^{2} - x$ in the equation ${(x - 1)}^{2} + {(y - 3)}^{2} - x = 0$ .

${(x - \frac{3}{2})}^{2} - \frac{5}{4} + {(y - 3)}^{2} = 0$

Step 4

Move $- \frac{5}{4}$ to the right side of the equation by adding $\frac{5}{4}$ to both sides.

${(x - \frac{3}{2})}^{2} + {(y - 3)}^{2} = 0 + \frac{5}{4}$

Step 5

Add $0$ and $\frac{5}{4}$ .

${(x - \frac{3}{2})}^{2} + {(y - 3)}^{2} = \frac{5}{4}$

Step 6

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 7

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.

$r = \frac{\sqrt{5}}{2}$

$h = \frac{3}{2}$

$k = 3$

Step 8

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

Center: $(\frac{3}{2}, 3)$

Step 9

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

Center: $(\frac{3}{2}, 3)$

Radius: $\frac{\sqrt{5}}{2}$

Step 10