Algebra Examples

$y = {(x + 3)}^{2} + {(x + 4)}^{2}$

Step 1

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

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite ${(x + 4)}^{2}$ as $(x + 4) (x + 4)$ .

$y = {(x + 3)}^{2} + (x + 4) (x + 4)$

Step 1.1.2

Expand $(x + 4) (x + 4)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$y = {(x + 3)}^{2} + x (x + 4) + 4 (x + 4)$

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

Move $4$ to the left of $x$ .

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

Step 1.1.3.1.3

Multiply $4$ by $4$ .

$y = {(x + 3)}^{2} + x^{2} + 4 x + 4 x + 16$

Step 1.1.3.2

Add $4 x$ and $4 x$ .

$y = {(x + 3)}^{2} + x^{2} + 8 x + 16$

Step 1.2

Reorder terms.

$y = x^{2} + {(x + 3)}^{2} + 8 x + 16$

Step 2

Complete the square for $x^{2} + {(x + 3)}^{2} + 8 x + 16$ .

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Step 2.1

Simplify the expression.

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$x^{2} + (x + 3) (x + 3) + 8 x + 16$

Step 2.1.1.2

Expand $(x + 3) (x + 3)$ using the FOIL Method.

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Step 2.1.1.2.1

Apply the distributive property.

$x^{2} + x (x + 3) + 3 (x + 3) + 8 x + 16$

Step 2.1.1.2.2

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 3 + 3 (x + 3) + 8 x + 16$

Step 2.1.1.2.3

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 + 8 x + 16$

Step 2.1.1.3

Simplify and combine like terms.

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Step 2.1.1.3.1

Simplify each term.

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Step 2.1.1.3.1.1

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 + 8 x + 16$

Step 2.1.1.3.1.2

Move $3$ to the left of $x$ .

$x^{2} + x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 + 8 x + 16$

Step 2.1.1.3.1.3

Multiply $3$ by $3$ .

$x^{2} + x^{2} + 3 x + 3 x + 9 + 8 x + 16$

Step 2.1.1.3.2

Add $3 x$ and $3 x$ .

$x^{2} + x^{2} + 6 x + 9 + 8 x + 16$

Step 2.1.2

Add $x^{2}$ and $x^{2}$ .

$2 x^{2} + 6 x + 9 + 8 x + 16$

Step 2.1.3

Add $6 x$ and $8 x$ .

$2 x^{2} + 14 x + 9 + 16$

Step 2.1.4

Add $9$ and $16$ .

$2 x^{2} + 14 x + 25$

Step 2.2

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

$a = 2$

$b = 14$

$c = 25$

Step 2.3

Consider the vertex form of a parabola.

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

Step 2.4

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

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Step 2.4.1

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

$d = \frac{14}{2 \cdot 2}$

Step 2.4.2

Cancel the common factor of $14$ and $2$ .

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Step 2.4.2.1

Factor $2$ out of $14$ .

$d = \frac{2 \cdot 7}{2 \cdot 2}$

Step 2.4.2.2

Cancel the common factors.

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Step 2.4.2.2.1

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

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

Step 2.4.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 7}{2 \cdot 2}$

Step 2.4.2.2.3

Rewrite the expression.

$d = \frac{7}{2}$

Step 2.5

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

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Step 2.5.1

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

$e = 25 - \frac{14^{2}}{4 \cdot 2}$

Step 2.5.2

Simplify the right side.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Raise $14$ to the power of $2$ .

$e = 25 - \frac{196}{4 \cdot 2}$

Step 2.5.2.1.2

Multiply $4$ by $2$ .

$e = 25 - \frac{196}{8}$

Step 2.5.2.1.3

Cancel the common factor of $196$ and $8$ .

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Step 2.5.2.1.3.1

Factor $4$ out of $196$ .

$e = 25 - \frac{4 (49)}{8}$

Step 2.5.2.1.3.2

Cancel the common factors.

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Step 2.5.2.1.3.2.1

Factor $4$ out of $8$ .

$e = 25 - \frac{4 \cdot 49}{4 \cdot 2}$

Step 2.5.2.1.3.2.2

Cancel the common factor.

$e = 25 - \frac{4 \cdot 49}{4 \cdot 2}$

Step 2.5.2.1.3.2.3

Rewrite the expression.

$e = 25 - \frac{49}{2}$

Step 2.5.2.2

To write $25$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$e = 25 \cdot \frac{2}{2} - \frac{49}{2}$

Step 2.5.2.3

Combine $25$ and $\frac{2}{2}$ .

$e = \frac{25 \cdot 2}{2} - \frac{49}{2}$

Step 2.5.2.4

Combine the numerators over the common denominator.

$e = \frac{25 \cdot 2 - 49}{2}$

Step 2.5.2.5

Simplify the numerator.

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Step 2.5.2.5.1

Multiply $25$ by $2$ .

$e = \frac{50 - 49}{2}$

Step 2.5.2.5.2

Subtract $49$ from $50$ .

$e = \frac{1}{2}$

Step 2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $2 {(x + \frac{7}{2})}^{2} + \frac{1}{2}$ .

$2 {(x + \frac{7}{2})}^{2} + \frac{1}{2}$

Step 3

Set $y$ equal to the new right side.

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