Find the Focus (3x-2)^2=84y-112

${(3 x - 2)}^{2} = 84 y - 112$

Step 1

Rewrite the equation in vertex form.

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

Rewrite the equation as $84 y - 112 = {(3 x - 2)}^{2}$ .

$84 y - 112 = {(3 x - 2)}^{2}$

Add $112$ to both sides of the equation.

$84 y = {(3 x - 2)}^{2} + 112$

Divide each term in $84 y = {(3 x - 2)}^{2} + 112$ by $84$ and simplify.

Divide each term in $84 y = {(3 x - 2)}^{2} + 112$ by $84$ .

$\frac{84 y}{84} = \frac{{(3 x - 2)}^{2}}{84} + \frac{112}{84}$

Simplify the left side.

Cancel the common factor of $84$ .

Cancel the common factor.

$\frac{84 y}{84} = \frac{{(3 x - 2)}^{2}}{84} + \frac{112}{84}$

Divide $y$ by $1$ .

$y = \frac{{(3 x - 2)}^{2}}{84} + \frac{112}{84}$

Simplify the right side.

Cancel the common factor of $112$ and $84$ .

Factor $28$ out of $112$ .

$y = \frac{{(3 x - 2)}^{2}}{84} + \frac{28 (4)}{84}$

Cancel the common factors.

Factor $28$ out of $84$ .

$y = \frac{{(3 x - 2)}^{2}}{84} + \frac{28 \cdot 4}{28 \cdot 3}$

Cancel the common factor.

$y = \frac{{(3 x - 2)}^{2}}{84} + \frac{28 \cdot 4}{28 \cdot 3}$

Rewrite the expression.

$y = \frac{{(3 x - 2)}^{2}}{84} + \frac{4}{3}$

Reorder terms.

$y = \frac{1}{84} \cdot {(3 x - 2)}^{2} + \frac{4}{3}$

Complete the square for $\frac{1}{84} \cdot {(3 x - 2)}^{2} + \frac{4}{3}$ .

Simplify the expression.

Simplify each term.

Rewrite ${(3 x - 2)}^{2}$ as $(3 x - 2) (3 x - 2)$ .

$\frac{1}{84} \cdot ((3 x - 2) (3 x - 2)) + \frac{4}{3}$

Expand $(3 x - 2) (3 x - 2)$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{84} \cdot (3 x (3 x - 2) - 2 (3 x - 2)) + \frac{4}{3}$

Apply the distributive property.

$\frac{1}{84} \cdot (3 x (3 x) + 3 x \cdot - 2 - 2 (3 x - 2)) + \frac{4}{3}$

Apply the distributive property.

$\frac{1}{84} \cdot (3 x (3 x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) + \frac{4}{3}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{1}{84} \cdot (3 \cdot 3 x \cdot x + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) + \frac{4}{3}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{1}{84} \cdot (3 \cdot 3 (x \cdot x) + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) + \frac{4}{3}$

Multiply $x$ by $x$ .

$\frac{1}{84} \cdot (3 \cdot 3 x^{2} + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) + \frac{4}{3}$

Multiply $3$ by $3$ .

$\frac{1}{84} \cdot (9 x^{2} + 3 x \cdot - 2 - 2 (3 x) - 2 \cdot - 2) + \frac{4}{3}$

Multiply $- 2$ by $3$ .

$\frac{1}{84} \cdot (9 x^{2} - 6 x - 2 (3 x) - 2 \cdot - 2) + \frac{4}{3}$

Multiply $3$ by $- 2$ .

$\frac{1}{84} \cdot (9 x^{2} - 6 x - 6 x - 2 \cdot - 2) + \frac{4}{3}$

Multiply $- 2$ by $- 2$ .

$\frac{1}{84} \cdot (9 x^{2} - 6 x - 6 x + 4) + \frac{4}{3}$

Subtract $6 x$ from $- 6 x$ .

$\frac{1}{84} \cdot (9 x^{2} - 12 x + 4) + \frac{4}{3}$

Apply the distributive property.

$\frac{1}{84} (9 x^{2}) + \frac{1}{84} (- 12 x) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Simplify.

Cancel the common factor of $3$ .

Factor $3$ out of $84$ .

$\frac{1}{3 (28)} (9 x^{2}) + \frac{1}{84} (- 12 x) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Factor $3$ out of $9 x^{2}$ .

$\frac{1}{3 (28)} (3 (3 x^{2})) + \frac{1}{84} (- 12 x) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Cancel the common factor.

$\frac{1}{3 \cdot 28} (3 (3 x^{2})) + \frac{1}{84} (- 12 x) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Rewrite the expression.

$\frac{1}{28} (3 x^{2}) + \frac{1}{84} (- 12 x) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Combine $3$ and $\frac{1}{28}$ .

$\frac{3}{28} x^{2} + \frac{1}{84} (- 12 x) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Combine $\frac{3}{28}$ and $x^{2}$ .

$\frac{3 x^{2}}{28} + \frac{1}{84} (- 12 x) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Cancel the common factor of $12$ .

Factor $12$ out of $84$ .

$\frac{3 x^{2}}{28} + \frac{1}{12 (7)} (- 12 x) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Factor $12$ out of $- 12 x$ .

$\frac{3 x^{2}}{28} + \frac{1}{12 (7)} (12 (- x)) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Cancel the common factor.

$\frac{3 x^{2}}{28} + \frac{1}{12 \cdot 7} (12 (- x)) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Rewrite the expression.

$\frac{3 x^{2}}{28} + \frac{1}{7} (- x) + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Combine $\frac{1}{7}$ and $x$ .

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{1}{84} \cdot 4 + \frac{4}{3}$

Cancel the common factor of $4$ .

Factor $4$ out of $84$ .

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{1}{4 (21)} \cdot 4 + \frac{4}{3}$

Cancel the common factor.

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{1}{4 \cdot 21} \cdot 4 + \frac{4}{3}$

Rewrite the expression.

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{1}{21} + \frac{4}{3}$

To write $\frac{4}{3}$ as a fraction with a common denominator, multiply by $\frac{7}{7}$ .

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{1}{21} + \frac{4}{3} \cdot \frac{7}{7}$

Write each expression with a common denominator of $21$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4}{3}$ by $\frac{7}{7}$ .

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{1}{21} + \frac{4 \cdot 7}{3 \cdot 7}$

Multiply $3$ by $7$ .

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{1}{21} + \frac{4 \cdot 7}{21}$

Combine the numerators over the common denominator.

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{1 + 4 \cdot 7}{21}$

Simplify the numerator.

Multiply $4$ by $7$ .

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{1 + 28}{21}$

Add $1$ and $28$ .

$\frac{3 x^{2}}{28} - \frac{x}{7} + \frac{29}{21}$

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

$a = \frac{3}{28}$

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

$c = \frac{29}{21}$

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

$d = \frac{- \frac{1}{7}}{2 (\frac{3}{28})}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{7} \cdot \frac{1}{2 (\frac{3}{28})}$

Combine $2$ and $\frac{3}{28}$ .

$d = - \frac{1}{7} \cdot \frac{1}{\frac{2 \cdot 3}{28}}$

Multiply $2$ by $3$ .

$d = - \frac{1}{7} \cdot \frac{1}{\frac{6}{28}}$

Cancel the common factor of $6$ and $28$ .

Factor $2$ out of $6$ .

$d = - \frac{1}{7} \cdot \frac{1}{\frac{2 (3)}{28}}$

Cancel the common factors.

Factor $2$ out of $28$ .

$d = - \frac{1}{7} \cdot \frac{1}{\frac{2 \cdot 3}{2 \cdot 14}}$

Cancel the common factor.

$d = - \frac{1}{7} \cdot \frac{1}{\frac{2 \cdot 3}{2 \cdot 14}}$

Rewrite the expression.

$d = - \frac{1}{7} \cdot \frac{1}{\frac{3}{14}}$

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{7} (1 (\frac{14}{3}))$

Multiply $\frac{14}{3}$ by $1$ .

$d = - \frac{1}{7} \cdot \frac{14}{3}$

Cancel the common factor of $7$ .

Move the leading negative in $- \frac{1}{7}$ into the numerator.

$d = \frac{- 1}{7} \cdot \frac{14}{3}$

Factor $7$ out of $14$ .

$d = \frac{- 1}{7} \cdot \frac{7 (2)}{3}$

Cancel the common factor.

$d = \frac{- 1}{7} \cdot \frac{7 \cdot 2}{3}$

Rewrite the expression.

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

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

$e = \frac{29}{21} - \frac{{(- \frac{1}{7})}^{2}}{4 (\frac{3}{28})}$

Simplify the right side.

Simplify each term.

Simplify the numerator.

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

$e = \frac{29}{21} - \frac{{(- 1)}^{2} {(\frac{1}{7})}^{2}}{4 (\frac{3}{28})}$

Raise $- 1$ to the power of $2$ .

$e = \frac{29}{21} - \frac{1 {(\frac{1}{7})}^{2}}{4 (\frac{3}{28})}$

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

$e = \frac{29}{21} - \frac{1 \frac{1^{2}}{7^{2}}}{4 (\frac{3}{28})}$

One to any power is one.

$e = \frac{29}{21} - \frac{1 \frac{1}{7^{2}}}{4 (\frac{3}{28})}$

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

$e = \frac{29}{21} - \frac{1 (\frac{1}{49})}{4 (\frac{3}{28})}$

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

$e = \frac{29}{21} - \frac{\frac{1}{49}}{4 (\frac{3}{28})}$

Combine $4$ and $\frac{3}{28}$ .

$e = \frac{29}{21} - \frac{\frac{1}{49}}{\frac{4 \cdot 3}{28}}$

Multiply $4$ by $3$ .

$e = \frac{29}{21} - \frac{\frac{1}{49}}{\frac{12}{28}}$

Cancel the common factor of $12$ and $28$ .

Factor $4$ out of $12$ .

$e = \frac{29}{21} - \frac{\frac{1}{49}}{\frac{4 (3)}{28}}$

Cancel the common factors.

Factor $4$ out of $28$ .

$e = \frac{29}{21} - \frac{\frac{1}{49}}{\frac{4 \cdot 3}{4 \cdot 7}}$

Cancel the common factor.

$e = \frac{29}{21} - \frac{\frac{1}{49}}{\frac{4 \cdot 3}{4 \cdot 7}}$

Rewrite the expression.

$e = \frac{29}{21} - \frac{\frac{1}{49}}{\frac{3}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{29}{21} - (\frac{1}{49} \cdot \frac{7}{3})$

Cancel the common factor of $7$ .

Factor $7$ out of $49$ .

$e = \frac{29}{21} - (\frac{1}{7 (7)} \cdot \frac{7}{3})$

Cancel the common factor.

$e = \frac{29}{21} - (\frac{1}{7 \cdot 7} \cdot \frac{7}{3})$

Rewrite the expression.

$e = \frac{29}{21} - (\frac{1}{7} \cdot \frac{1}{3})$

Multiply $\frac{1}{7}$ by $\frac{1}{3}$ .

$e = \frac{29}{21} - \frac{1}{7 \cdot 3}$

Multiply $7$ by $3$ .

$e = \frac{29}{21} - \frac{1}{21}$

Combine the numerators over the common denominator.

$e = \frac{29 - 1}{21}$

Subtract $1$ from $29$ .

$e = \frac{28}{21}$

Cancel the common factor of $28$ and $21$ .

Factor $7$ out of $28$ .

$e = \frac{7 (4)}{21}$

Cancel the common factors.

Factor $7$ out of $21$ .

$e = \frac{7 \cdot 4}{7 \cdot 3}$

Cancel the common factor.

$e = \frac{7 \cdot 4}{7 \cdot 3}$

Rewrite the expression.

$e = \frac{4}{3}$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{3}{28} {(x - \frac{2}{3})}^{2} + \frac{4}{3}$ .

$\frac{3}{28} {(x - \frac{2}{3})}^{2} + \frac{4}{3}$

Set $y$ equal to the new right side.

$y = \frac{3}{28} \cdot {(x - \frac{2}{3})}^{2} + \frac{4}{3}$

Step 2

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

$a = \frac{3}{28}$

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

$k = \frac{4}{3}$

Step 3

Find the vertex $(h, k)$ .

$(\frac{2}{3}, \frac{4}{3})$

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.

$\frac{1}{4 \cdot \frac{3}{28}}$

Simplify.

Combine $4$ and $\frac{3}{28}$ .

$\frac{1}{\frac{4 \cdot 3}{28}}$

Reduce the expression by cancelling the common factors.

Multiply $4$ by $3$ .

$\frac{1}{\frac{12}{28}}$

Cancel the common factor of $12$ and $28$ .

Factor $4$ out of $12$ .

$\frac{1}{\frac{4 (3)}{28}}$

Cancel the common factors.

Factor $4$ out of $28$ .

$\frac{1}{\frac{4 \cdot 3}{4 \cdot 7}}$

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 3}{4 \cdot 7}}$

Rewrite the expression.

$\frac{1}{\frac{3}{7}}$

Multiply the numerator by the reciprocal of the denominator.

$1 (\frac{7}{3})$

Multiply $\frac{7}{3}$ by $1$ .

$\frac{7}{3}$

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.

$(\frac{2}{3}, \frac{11}{3})$

Step 6

Find the Focus (3x-2)^2=84y-112

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