Algebra Examples

$y + 2 x + 3 = - {(x + 2)}^{2} + 1$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

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

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Step 1.1.1

Subtract $2 x$ from both sides of the equation.

$y + 3 = - {(x + 2)}^{2} + 1 - 2 x$

Step 1.1.2

Subtract $3$ from both sides of the equation.

$y = - {(x + 2)}^{2} + 1 - 2 x - 3$

Step 1.1.3

Subtract $3$ from $1$ .

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

Step 1.2

Complete the square for $- {(x + 2)}^{2} - 2 x - 2$ .

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Step 1.2.1

Simplify the expression.

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Rewrite ${(x + 2)}^{2}$ as $(x + 2) (x + 2)$ .

$- ((x + 2) (x + 2)) - 2 x - 2$

Step 1.2.1.1.2

Expand $(x + 2) (x + 2)$ using the FOIL Method.

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Step 1.2.1.1.2.1

Apply the distributive property.

$- (x (x + 2) + 2 (x + 2)) - 2 x - 2$

Step 1.2.1.1.2.2

Apply the distributive property.

$- (x \cdot x + x \cdot 2 + 2 (x + 2)) - 2 x - 2$

Step 1.2.1.1.2.3

Apply the distributive property.

$- (x \cdot x + x \cdot 2 + 2 x + 2 \cdot 2) - 2 x - 2$

Step 1.2.1.1.3

Simplify and combine like terms.

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Step 1.2.1.1.3.1

Simplify each term.

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Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$- (x^{2} + x \cdot 2 + 2 x + 2 \cdot 2) - 2 x - 2$

Step 1.2.1.1.3.1.2

Move $2$ to the left of $x$ .

$- (x^{2} + 2 \cdot x + 2 x + 2 \cdot 2) - 2 x - 2$

Step 1.2.1.1.3.1.3

Multiply $2$ by $2$ .

$- (x^{2} + 2 x + 2 x + 4) - 2 x - 2$

Step 1.2.1.1.3.2

Add $2 x$ and $2 x$ .

$- (x^{2} + 4 x + 4) - 2 x - 2$

Step 1.2.1.1.4

Apply the distributive property.

$- x^{2} - (4 x) - 1 \cdot 4 - 2 x - 2$

Step 1.2.1.1.5

Simplify.

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Step 1.2.1.1.5.1

Multiply $4$ by $- 1$ .

$- x^{2} - 4 x - 1 \cdot 4 - 2 x - 2$

Step 1.2.1.1.5.2

Multiply $- 1$ by $4$ .

$- x^{2} - 4 x - 4 - 2 x - 2$

Step 1.2.1.2

Subtract $2 x$ from $- 4 x$ .

$- x^{2} - 6 x - 4 - 2$

Step 1.2.1.3

Subtract $2$ from $- 4$ .

$- x^{2} - 6 x - 6$

Step 1.2.2

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

$a = - 1$

$b = - 6$

$c = - 6$

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

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Step 1.2.4.1

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

$d = \frac{- 6}{2 \cdot - 1}$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Cancel the common factor of $- 6$ and $2$ .

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Step 1.2.4.2.1.1

Factor $2$ out of $- 6$ .

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

Step 1.2.4.2.1.2

Move the negative one from the denominator of $\frac{- 3}{- 1}$ .

$d = - 1 \cdot - 3$

Step 1.2.4.2.2

Rewrite $- 1 \cdot - 3$ as $- - 3$ .

$d = - - 3$

Step 1.2.4.2.3

Multiply $- 1$ by $- 3$ .

$d = 3$

Step 1.2.5

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

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Step 1.2.5.1

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

$e = - 6 - \frac{{(- 6)}^{2}}{4 \cdot - 1}$

Step 1.2.5.2

Simplify the right side.

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Step 1.2.5.2.1

Simplify each term.

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Step 1.2.5.2.1.1

Raise $- 6$ to the power of $2$ .

$e = - 6 - \frac{36}{4 \cdot - 1}$

Step 1.2.5.2.1.2

Multiply $4$ by $- 1$ .

$e = - 6 - \frac{36}{- 4}$

Step 1.2.5.2.1.3

Divide $36$ by $- 4$ .

$e = - 6 - - 9$

Step 1.2.5.2.1.4

Multiply $- 1$ by $- 9$ .

$e = - 6 + 9$

Step 1.2.5.2.2

Add $- 6$ and $9$ .

$e = 3$

Step 1.2.6

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

$- {(x + 3)}^{2} + 3$

Step 1.3

Set $y$ equal to the new right side.

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

Step 2

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

$a = - 1$

$h = - 3$

$k = 3$

Step 3

Find the vertex $(h, k)$ .

$(- 3, 3)$

Step 4