Precalculus Examples

$- x^{2} + 10 x + y - 21 = 0$

Step 1

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 1.1

Subtract $y$ from both sides of the equation.

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

Step 1.2

Add $21$ to both sides of the equation.

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

Step 2

Complete the square.

Tap for more steps...

Step 2.1

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

$a = - 1$

$b = 10$

$c = 0 = - y + 21$

Step 2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = - y + 21$

Step 2.3

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

Tap for more steps...

Step 2.3.1

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

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

Step 2.3.2

Simplify the right side.

Tap for more steps...

Step 2.3.2.1

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

Tap for more steps...

Step 2.3.2.1.1

Factor $2$ out of $10$ .

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

Step 2.3.2.1.2

Move the negative one from the denominator of $\frac{5}{- 1}$ .

$d = - 1 \cdot 5$

Step 2.3.2.2

Multiply $- 1$ by $5$ .

$d = - 5$

Step 2.4

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

Tap for more steps...

Step 2.4.1

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

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

Step 2.4.2

Simplify the right side.

Tap for more steps...

Step 2.4.2.1

Simplify each term.

Tap for more steps...

Step 2.4.2.1.1

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

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

Step 2.4.2.1.2

Multiply $4$ by $- 1$ .

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

Step 2.4.2.1.3

Divide $100$ by $- 4$ .

$e = 0 - - 25$

Step 2.4.2.1.4

Multiply $- 1$ by $- 25$ .

$e = 0 + 25$

Step 2.4.2.2

Add $0$ and $25$ .

$e = 25$

Step 2.5

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

$- {(x - 5)}^{2} + 25 = - y + 21$

Step 3

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 3.1

Subtract $25$ from both sides of the equation.

$- {(x - 5)}^{2} = - y + 21 - 25$

Step 3.2

Subtract $25$ from $21$ .

$- {(x - 5)}^{2} = - y - 4$

Step 4

Divide each term in $- {(x - 5)}^{2} = - y - 4$ by $- 1$ and simplify.

Tap for more steps...

Step 4.1

Divide each term in $- {(x - 5)}^{2} = - y - 4$ by $- 1$ .

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

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Dividing two negative values results in a positive value.

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

Step 4.2.2

Divide ${(x - 5)}^{2}$ by $1$ .

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

Step 4.3

Simplify the right side.

Tap for more steps...

Step 4.3.1

Simplify each term.

Tap for more steps...

Step 4.3.1.1

Dividing two negative values results in a positive value.

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

Step 4.3.1.2

Divide $y$ by $1$ .

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

Step 4.3.1.3

Divide $- 4$ by $- 1$ .

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