Algebra Examples

$x^{2} + 2 x + y^{2} = 55 x + 10 y$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Subtract $55 x$ from both sides of the equation.

$x^{2} + 2 x + y^{2} - 55 x = 10 y$

Step 1.2

Subtract $10 y$ from both sides of the equation.

$x^{2} + 2 x + y^{2} - 55 x - 10 y = 0$

Step 1.3

Subtract $55 x$ from $2 x$ .

$x^{2} + y^{2} - 53 x - 10 y = 0$

Step 2

Complete the square for $x^{2} - 53 x$ .

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Step 2.1

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

$a = 1$

$b = - 53$

$c = 0$

Step 2.2

Consider the vertex form of a parabola.

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

Step 2.3

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

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Step 2.3.1

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

$d = \frac{- 53}{2 \cdot 1}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $2$ by $1$ .

$d = \frac{- 53}{2}$

Step 2.3.2.2

Move the negative in front of the fraction.

$d = - \frac{53}{2}$

Step 2.4

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

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Step 2.4.1

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

$e = 0 - \frac{{(- 53)}^{2}}{4 \cdot 1}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $- 53$ to the power of $2$ .

$e = 0 - \frac{2809}{4 \cdot 1}$

Step 2.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{2809}{4}$

Step 2.4.2.2

Subtract $\frac{2809}{4}$ from $0$ .

$e = - \frac{2809}{4}$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - \frac{53}{2})}^{2} - \frac{2809}{4}$ .

${(x - \frac{53}{2})}^{2} - \frac{2809}{4}$

Step 3

Substitute ${(x - \frac{53}{2})}^{2} - \frac{2809}{4}$ for $x^{2} - 53 x$ in the equation $x^{2} + y^{2} - 53 x - 10 y = 0$ .

${(x - \frac{53}{2})}^{2} - \frac{2809}{4} + y^{2} - 10 y = 0$

Step 4

Move $- \frac{2809}{4}$ to the right side of the equation by adding $\frac{2809}{4}$ to both sides.

${(x - \frac{53}{2})}^{2} + y^{2} - 10 y = 0 + \frac{2809}{4}$

Step 5

Complete the square for $y^{2} - 10 y$ .

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Step 5.1

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

$a = 1$

$b = - 10$

$c = 0$

Step 5.2

Consider the vertex form of a parabola.

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

Step 5.3

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

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Step 5.3.1

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

$d = \frac{- 10}{2 \cdot 1}$

Step 5.3.2

Cancel the common factor of $- 10$ and $2$ .

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Step 5.3.2.1

Factor $2$ out of $- 10$ .

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Step 5.3.2.2

Cancel the common factors.

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Step 5.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 5}{2 (1)}$

Step 5.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 5}{2 \cdot 1}$

Step 5.3.2.2.3

Rewrite the expression.

$d = \frac{- 5}{1}$

Step 5.3.2.2.4

Divide $- 5$ by $1$ .

$d = - 5$

Step 5.4

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

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Step 5.4.1

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

$e = 0 - \frac{{(- 10)}^{2}}{4 \cdot 1}$

Step 5.4.2

Simplify the right side.

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Step 5.4.2.1

Simplify each term.

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Step 5.4.2.1.1

Raise $- 10$ to the power of $2$ .

$e = 0 - \frac{100}{4 \cdot 1}$

Step 5.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{100}{4}$

Step 5.4.2.1.3

Divide $100$ by $4$ .

$e = 0 - 1 \cdot 25$

Step 5.4.2.1.4

Multiply $- 1$ by $25$ .

$e = 0 - 25$

Step 5.4.2.2

Subtract $25$ from $0$ .

$e = - 25$

Step 5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 5)}^{2} - 25$ .

${(y - 5)}^{2} - 25$

Step 6

Substitute ${(y - 5)}^{2} - 25$ for $y^{2} - 10 y$ in the equation $x^{2} + y^{2} - 53 x - 10 y = 0$ .

${(x - \frac{53}{2})}^{2} + {(y - 5)}^{2} - 25 = 0 + \frac{2809}{4}$

Step 7

Move $- 25$ to the right side of the equation by adding $25$ to both sides.

${(x - \frac{53}{2})}^{2} + {(y - 5)}^{2} = 0 + \frac{2809}{4} + 25$

Step 8

Simplify $0 + \frac{2809}{4} + 25$ .

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Step 8.1

Add $0$ and $\frac{2809}{4}$ .

${(x - \frac{53}{2})}^{2} + {(y - 5)}^{2} = \frac{2809}{4} + 25$

Step 8.2

To write $25$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(x - \frac{53}{2})}^{2} + {(y - 5)}^{2} = \frac{2809}{4} + 25 \cdot \frac{4}{4}$

Step 8.3

Combine $25$ and $\frac{4}{4}$ .

${(x - \frac{53}{2})}^{2} + {(y - 5)}^{2} = \frac{2809}{4} + \frac{25 \cdot 4}{4}$

Step 8.4

Combine the numerators over the common denominator.

${(x - \frac{53}{2})}^{2} + {(y - 5)}^{2} = \frac{2809 + 25 \cdot 4}{4}$

Step 8.5

Simplify the numerator.

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Step 8.5.1

Multiply $25$ by $4$ .

${(x - \frac{53}{2})}^{2} + {(y - 5)}^{2} = \frac{2809 + 100}{4}$

Step 8.5.2

Add $2809$ and $100$ .

${(x - \frac{53}{2})}^{2} + {(y - 5)}^{2} = \frac{2909}{4}$