Algebra Examples

${(x + 3)}^{2} - 5$

Step 1

Simplify each term.

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Step 1.1

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

$f (x) = (x + 3) (x + 3) - 5$

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

$f (x) = x (x + 3) + 3 (x + 3) - 5$

Step 1.2.2

Apply the distributive property.

$f (x) = x \cdot x + x \cdot 3 + 3 (x + 3) - 5$

Step 1.2.3

Apply the distributive property.

$f (x) = x \cdot x + x \cdot 3 + 3 x + 3 \cdot 3 - 5$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Multiply $x$ by $x$ .

$f (x) = x^{2} + x \cdot 3 + 3 x + 3 \cdot 3 - 5$

Step 1.3.1.2

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

$f (x) = x^{2} + 3 \cdot x + 3 x + 3 \cdot 3 - 5$

Step 1.3.1.3

Multiply $3$ by $3$ .

$f (x) = x^{2} + 3 x + 3 x + 9 - 5$

Step 1.3.2

Add $3 x$ and $3 x$ .

$f (x) = x^{2} + 6 x + 9 - 5$

Step 2

Subtract $5$ from $9$ .

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

Step 3

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 4

Find the value of $x = - \frac{b}{2 a}$ .

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Step 4.1

Substitute in the values of $a$ and $b$ .

$x = - \frac{6}{2 (1)}$

Step 4.2

Remove parentheses.

$x = - \frac{6}{2 (1)}$

Step 4.3

Simplify $- \frac{6}{2 (1)}$ .

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Step 4.3.1

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

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Step 4.3.1.1

Factor $2$ out of $6$ .

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

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $2$ out of $2 \cdot 1$ .

$x = - \frac{2 \cdot 3}{2 (1)}$

Step 4.3.1.2.2

Cancel the common factor.

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

Step 4.3.1.2.3

Rewrite the expression.

$x = - \frac{3}{1}$

Step 4.3.1.2.4

Divide $3$ by $1$ .

$x = - 1 \cdot 3$

Step 4.3.2

Multiply $- 1$ by $3$ .

$x = - 3$

Step 5

Evaluate $f (- 3)$ .

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Step 5.1

Replace the variable $x$ with $- 3$ in the expression.

$f (- 3) = {(- 3)}^{2} + 6 (- 3) + 4$

Step 5.2

Simplify the result.

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Step 5.2.1

Simplify each term.

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Step 5.2.1.1

Raise $- 3$ to the power of $2$ .

$f (- 3) = 9 + 6 (- 3) + 4$

Step 5.2.1.2

Multiply $6$ by $- 3$ .

$f (- 3) = 9 - 18 + 4$

Step 5.2.2

Simplify by adding and subtracting.

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Step 5.2.2.1

Subtract $18$ from $9$ .

$f (- 3) = - 9 + 4$

Step 5.2.2.2

Add $- 9$ and $4$ .

$f (- 3) = - 5$

Step 5.2.3

The final answer is $- 5$ .

$- 5$

Step 6

Use the $x$ and $y$ values to find where the minimum occurs.

$(- 3, - 5)$

Step 7