Precalculus Examples

$x^{2} + 2 y = 8 x - 7$

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

Subtract $x^{2}$ from both sides of the equation.

$2 y = 8 x - 7 - x^{2}$

Step 1.1.2

Divide each term in $2 y = 8 x - 7 - x^{2}$ by $2$ and simplify.

Tap for more steps...

Step 1.1.2.1

Divide each term in $2 y = 8 x - 7 - x^{2}$ by $2$ .

$\frac{2 y}{2} = \frac{8 x}{2} + \frac{- 7}{2} + \frac{- x^{2}}{2}$

Step 1.1.2.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{8 x}{2} + \frac{- 7}{2} + \frac{- x^{2}}{2}$

Step 1.1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{8 x}{2} + \frac{- 7}{2} + \frac{- x^{2}}{2}$

Step 1.1.2.3

Simplify the right side.

Tap for more steps...

Step 1.1.2.3.1

Simplify each term.

Tap for more steps...

Step 1.1.2.3.1.1

Cancel the common factor of $8$ and $2$ .

Tap for more steps...

Step 1.1.2.3.1.1.1

Factor $2$ out of $8 x$ .

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

Step 1.1.2.3.1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.1.2.3.1.1.2.1

Factor $2$ out of $2$ .

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

Step 1.1.2.3.1.1.2.2

Cancel the common factor.

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

Step 1.1.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{4 x}{1} + \frac{- 7}{2} + \frac{- x^{2}}{2}$

Step 1.1.2.3.1.1.2.4

Divide $4 x$ by $1$ .

$y = 4 x + \frac{- 7}{2} + \frac{- x^{2}}{2}$

Step 1.1.2.3.1.2

Move the negative in front of the fraction.

$y = 4 x - \frac{7}{2} + \frac{- x^{2}}{2}$

Step 1.1.2.3.1.3

Move the negative in front of the fraction.

$y = 4 x - \frac{7}{2} - \frac{x^{2}}{2}$

Step 1.2

Complete the square for $4 x - \frac{7}{2} - \frac{x^{2}}{2}$ .

Tap for more steps...

Step 1.2.1

Simplify the expression.

Tap for more steps...

Step 1.2.1.1

Move $- \frac{7}{2}$ .

$4 x - \frac{x^{2}}{2} - \frac{7}{2}$

Step 1.2.1.2

Reorder $4 x$ and $- \frac{x^{2}}{2}$ .

$- \frac{x^{2}}{2} + 4 x - \frac{7}{2}$

Step 1.2.2

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

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

$b = 4$

$c = - \frac{7}{2}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

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

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

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

Tap for more steps...

Step 1.2.4.2.1.1

Factor $2$ out of $4$ .

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

Step 1.2.4.2.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.4.2.1.2.1

Cancel the common factor.

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

Step 1.2.4.2.1.2.2

Rewrite the expression.

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

Step 1.2.4.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 2 (- 1 \cdot 2)$

Step 1.2.4.2.3

Multiply $2 (- 1 \cdot 2)$ .

Tap for more steps...

Step 1.2.4.2.3.1

Multiply $- 1$ by $2$ .

$d = 2 \cdot - 2$

Step 1.2.4.2.3.2

Multiply $2$ by $- 2$ .

$d = - 4$

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

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

Step 1.2.5.2

Simplify the right side.

Tap for more steps...

Step 1.2.5.2.1

Simplify each term.

Tap for more steps...

Step 1.2.5.2.1.1

Cancel the common factor of $4^{2}$ and $4$ .

Tap for more steps...

Step 1.2.5.2.1.1.1

Factor $4$ out of $4^{2}$ .

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

Step 1.2.5.2.1.1.2

Cancel the common factors.

Tap for more steps...

Step 1.2.5.2.1.1.2.1

Cancel the common factor.

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

Step 1.2.5.2.1.1.2.2

Rewrite the expression.

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

Step 1.2.5.2.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.2.5.2.1.3

Multiply $- (4 (- 1 \cdot 2))$ .

Tap for more steps...

Step 1.2.5.2.1.3.1

Multiply $- 1$ by $2$ .

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

Step 1.2.5.2.1.3.2

Multiply $4$ by $- 2$ .

$e = - \frac{7}{2} - - 8$

Step 1.2.5.2.1.3.3

Multiply $- 1$ by $- 8$ .

$e = - \frac{7}{2} + 8$

Step 1.2.5.2.2

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = - \frac{7}{2} + 8 \cdot \frac{2}{2}$

Step 1.2.5.2.3

Combine $8$ and $\frac{2}{2}$ .

$e = - \frac{7}{2} + \frac{8 \cdot 2}{2}$

Step 1.2.5.2.4

Combine the numerators over the common denominator.

$e = \frac{- 7 + 8 \cdot 2}{2}$

Step 1.2.5.2.5

Simplify the numerator.

Tap for more steps...

Step 1.2.5.2.5.1

Multiply $8$ by $2$ .

$e = \frac{- 7 + 16}{2}$

Step 1.2.5.2.5.2

Add $- 7$ and $16$ .

$e = \frac{9}{2}$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{1}{2} {(x - 4)}^{2} + \frac{9}{2}$ .

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

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

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

$h = 4$

$k = \frac{9}{2}$

Step 3

Find the vertex $(h, k)$ .

$(4, \frac{9}{2})$

Step 4