Algebra Examples

$f (x) = - 5 x^{2} - 6 - 4 x$

Step 1

Write $f (x) = - 5 x^{2} - 6 - 4 x$ as an equation.

$y = - 5 x^{2} - 6 - 4 x$

Step 2

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$- 5 x^{2} - 6 - 4 x = y$

Step 3

Add $6$ to both sides of the equation.

$- 5 x^{2} - 4 x = y + 6$

Step 4

Complete the square.

Tap for more steps...

Step 4.1

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

$a = - 5$

$b = - 4$

$c = 0 = y + 6$

Step 4.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e = y + 6$

Step 4.3

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

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

Simplify the right side.

Tap for more steps...

Step 4.3.2.1

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

Tap for more steps...

Step 4.3.2.1.1

Factor $2$ out of $- 4$ .

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

Step 4.3.2.1.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.1.2.1

Factor $2$ out of $2 \cdot - 5$ .

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

Step 4.3.2.1.2.2

Cancel the common factor.

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

Step 4.3.2.1.2.3

Rewrite the expression.

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

Step 4.3.2.2

Dividing two negative values results in a positive value.

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

Step 4.4

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

Tap for more steps...

Step 4.4.1

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

$e = 0 - \frac{{(- 4)}^{2}}{4 \cdot - 5}$

Step 4.4.2

Simplify the right side.

Tap for more steps...

Step 4.4.2.1

Simplify each term.

Tap for more steps...

Step 4.4.2.1.1

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

Tap for more steps...

Step 4.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 \cdot - 5}$

Step 4.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot - 5}$

Step 4.4.2.1.1.3

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

$e = 0 - \frac{1 \cdot 4^{2}}{4 \cdot - 5}$

Step 4.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = 0 - \frac{4^{2}}{4 \cdot - 5}$

Step 4.4.2.1.1.5

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

$e = 0 - \frac{4 \cdot 4}{4 \cdot - 5}$

Step 4.4.2.1.1.6

Cancel the common factors.

Tap for more steps...

Step 4.4.2.1.1.6.1

Factor $4$ out of $4 \cdot - 5$ .

$e = 0 - \frac{4 \cdot 4}{4 (- 5)}$

Step 4.4.2.1.1.6.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 \cdot - 5}$

Step 4.4.2.1.1.6.3

Rewrite the expression.

$e = 0 - \frac{4}{- 5}$

Step 4.4.2.1.2

Move the negative in front of the fraction.

$e = 0 - - \frac{4}{5}$

Step 4.4.2.1.3

Multiply $- - \frac{4}{5}$ .

Tap for more steps...

Step 4.4.2.1.3.1

Multiply $- 1$ by $- 1$ .

$e = 0 + 1 (\frac{4}{5})$

Step 4.4.2.1.3.2

Multiply $\frac{4}{5}$ by $1$ .

$e = 0 + \frac{4}{5}$

Step 4.4.2.2

Add $0$ and $\frac{4}{5}$ .

$e = \frac{4}{5}$

Step 4.5

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

$- 5 {(x + \frac{2}{5})}^{2} + \frac{4}{5} = y + 6$

Step 5

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

Tap for more steps...

Step 5.1

Subtract $\frac{4}{5}$ from both sides of the equation.

$- 5 {(x + \frac{2}{5})}^{2} = y + 6 - \frac{4}{5}$

Step 5.2

To write $6$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- 5 {(x + \frac{2}{5})}^{2} = y + 6 \cdot \frac{5}{5} - \frac{4}{5}$

Step 5.3

Combine $6$ and $\frac{5}{5}$ .

$- 5 {(x + \frac{2}{5})}^{2} = y + \frac{6 \cdot 5}{5} - \frac{4}{5}$

Step 5.4

Combine the numerators over the common denominator.

$- 5 {(x + \frac{2}{5})}^{2} = y + \frac{6 \cdot 5 - 4}{5}$

Step 5.5

Simplify the numerator.

Tap for more steps...

Step 5.5.1

Multiply $6$ by $5$ .

$- 5 {(x + \frac{2}{5})}^{2} = y + \frac{30 - 4}{5}$

Step 5.5.2

Subtract $4$ from $30$ .

$- 5 {(x + \frac{2}{5})}^{2} = y + \frac{26}{5}$

Step 6

Divide each term in $- 5 {(x + \frac{2}{5})}^{2} = y + \frac{26}{5}$ by $- 5$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $- 5 {(x + \frac{2}{5})}^{2} = y + \frac{26}{5}$ by $- 5$ .

$\frac{- 5 {(x + \frac{2}{5})}^{2}}{- 5} = \frac{y}{- 5} + \frac{\frac{26}{5}}{- 5}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $- 5$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{- 5 {(x + \frac{2}{5})}^{2}}{- 5} = \frac{y}{- 5} + \frac{\frac{26}{5}}{- 5}$

Step 6.2.1.2

Divide ${(x + \frac{2}{5})}^{2}$ by $1$ .

${(x + \frac{2}{5})}^{2} = \frac{y}{- 5} + \frac{\frac{26}{5}}{- 5}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Simplify each term.

Tap for more steps...

Step 6.3.1.1

Move the negative in front of the fraction.

${(x + \frac{2}{5})}^{2} = - \frac{y}{5} + \frac{\frac{26}{5}}{- 5}$

Step 6.3.1.2

Multiply the numerator by the reciprocal of the denominator.

${(x + \frac{2}{5})}^{2} = - \frac{y}{5} + \frac{26}{5} \cdot \frac{1}{- 5}$

Step 6.3.1.3

Move the negative in front of the fraction.

${(x + \frac{2}{5})}^{2} = - \frac{y}{5} + \frac{26}{5} (- \frac{1}{5})$

Step 6.3.1.4

Multiply $\frac{26}{5} (- \frac{1}{5})$ .

Tap for more steps...

Step 6.3.1.4.1

Multiply $\frac{26}{5}$ by $\frac{1}{5}$ .

${(x + \frac{2}{5})}^{2} = - \frac{y}{5} - \frac{26}{5 \cdot 5}$

Step 6.3.1.4.2

Multiply $5$ by $5$ .

${(x + \frac{2}{5})}^{2} = - \frac{y}{5} - \frac{26}{25}$

Step 7

Factor.

Tap for more steps...

Step 7.1

Factor $- 1$ out of $- \frac{y}{5} - \frac{26}{25}$ .

Tap for more steps...

Step 7.1.1

Factor $- 1$ out of $- \frac{y}{5}$ .

${(x + \frac{2}{5})}^{2} = - (\frac{y}{5}) - \frac{26}{25}$

Step 7.1.2

Factor $- 1$ out of $- \frac{26}{25}$ .

${(x + \frac{2}{5})}^{2} = - (\frac{y}{5}) - (\frac{26}{25})$

Step 7.1.3

Factor $- 1$ out of $- (\frac{y}{5}) - (\frac{26}{25})$ .

${(x + \frac{2}{5})}^{2} = - (\frac{y}{5} + \frac{26}{25})$

Step 7.2

To write $\frac{y}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

${(x + \frac{2}{5})}^{2} = - (\frac{y}{5} \cdot \frac{5}{5} + \frac{26}{25})$

Step 7.3

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 7.3.1

Multiply $\frac{y}{5}$ by $\frac{5}{5}$ .

${(x + \frac{2}{5})}^{2} = - (\frac{y \cdot 5}{5 \cdot 5} + \frac{26}{25})$

Step 7.3.2

Multiply $5$ by $5$ .

${(x + \frac{2}{5})}^{2} = - (\frac{y \cdot 5}{25} + \frac{26}{25})$

Step 7.4

Combine the numerators over the common denominator.

${(x + \frac{2}{5})}^{2} = - \frac{y \cdot 5 + 26}{25}$

Step 7.5

Move $5$ to the left of $y$ .

${(x + \frac{2}{5})}^{2} = - \frac{5 y + 26}{25}$

Step 7.6

Reorder terms.

${(x + \frac{2}{5})}^{2} = - (\frac{1}{25} (5 y + 26))$

Step 7.7

Remove parentheses.

${(x + \frac{2}{5})}^{2} = - \frac{1}{25} (5 y + 26)$