Find the Focus x^2+10x=4y-21

$x^{2} + 10 x = 4 y - 21$

Step 1

Rewrite the equation in vertex form.

Isolate $y$ to the left side of the equation.

Rewrite the equation as $4 y - 21 = x^{2} + 10 x$ .

$4 y - 21 = x^{2} + 10 x$

Add $21$ to both sides of the equation.

$4 y = x^{2} + 10 x + 21$

Divide each term in $4 y = x^{2} + 10 x + 21$ by $4$ and simplify.

Divide each term in $4 y = x^{2} + 10 x + 21$ by $4$ .

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{10 x}{4} + \frac{21}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 y}{4} = \frac{x^{2}}{4} + \frac{10 x}{4} + \frac{21}{4}$

Divide $y$ by $1$ .

$y = \frac{x^{2}}{4} + \frac{10 x}{4} + \frac{21}{4}$

Simplify the right side.

Cancel the common factor of $10$ and $4$ .

Factor $2$ out of $10 x$ .

$y = \frac{x^{2}}{4} + \frac{2 (5 x)}{4} + \frac{21}{4}$

Cancel the common factors.

Factor $2$ out of $4$ .

$y = \frac{x^{2}}{4} + \frac{2 (5 x)}{2 (2)} + \frac{21}{4}$

Cancel the common factor.

$y = \frac{x^{2}}{4} + \frac{2 (5 x)}{2 \cdot 2} + \frac{21}{4}$

Rewrite the expression.

$y = \frac{x^{2}}{4} + \frac{5 x}{2} + \frac{21}{4}$

Complete the square for $\frac{x^{2}}{4} + \frac{5 x}{2} + \frac{21}{4}$ .

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

$a = \frac{1}{4}$

$b = \frac{5}{2}$

$c = \frac{21}{4}$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{\frac{5}{2}}{2 (\frac{1}{4})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = \frac{5}{2} \cdot \frac{1}{2 (\frac{1}{4})}$

Combine $2$ and $\frac{1}{4}$ .

$d = \frac{5}{2} \cdot \frac{1}{\frac{2}{4}}$

Cancel the common factor of $2$ and $4$ .

Factor $2$ out of $2$ .

$d = \frac{5}{2} \cdot \frac{1}{\frac{2 (1)}{4}}$

Cancel the common factors.

Factor $2$ out of $4$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $2$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

$d = 5$

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

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

$e = \frac{21}{4} - \frac{{(\frac{5}{2})}^{2}}{4 (\frac{1}{4})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

Apply the product rule to $\frac{5}{2}$ .

$e = \frac{21}{4} - \frac{\frac{5^{2}}{2^{2}}}{4 (\frac{1}{4})}$

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

$e = \frac{21}{4} - \frac{\frac{25}{2^{2}}}{4 (\frac{1}{4})}$

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

$e = \frac{21}{4} - \frac{\frac{25}{4}}{4 (\frac{1}{4})}$

Combine $4$ and $\frac{1}{4}$ .

$e = \frac{21}{4} - \frac{\frac{25}{4}}{\frac{4}{4}}$

Divide $4$ by $4$ .

$e = \frac{21}{4} - \frac{\frac{25}{4}}{1}$

Divide $\frac{25}{4}$ by $1$ .

$e = \frac{21}{4} - \frac{25}{4}$

Combine the numerators over the common denominator.

$e = \frac{21 - 25}{4}$

Subtract $25$ from $21$ .

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

Divide $- 4$ by $4$ .

$e = - 1$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{4} {(x + 5)}^{2} - 1$ .

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

Set $y$ equal to the new right side.

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

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = \frac{1}{4}$

$h = - 5$

$k = - 1$

Step 3

Find the vertex $(h, k)$ .

$(- 5, - 1)$

Step 4

Find $p$ , the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot \frac{1}{4}}$

Simplify.

Combine $4$ and $\frac{1}{4}$ .

$\frac{1}{\frac{4}{4}}$

Simplify by dividing numbers.

Divide $4$ by $4$ .

$\frac{1}{1}$

Divide $1$ by $1$ .

$1$

Step 5

Find the focus.

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(- 5, 0)$

Step 6

Find the Focus x^2+10x=4y-21

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