Algebra Examples

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

Step 1

Simplify each term.

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Step 1.1

Rewrite ${(x + \frac{1}{3})}^{2}$ as $(x + \frac{1}{3}) (x + \frac{1}{3})$ .

$f (x) = 3 ((x + \frac{1}{3}) (x + \frac{1}{3})) + \frac{2}{3}$

Step 1.2

Expand $(x + \frac{1}{3}) (x + \frac{1}{3})$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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 $x$ .

$f (x) = 3 (x^{2} + x (\frac{1}{3}) + \frac{1}{3} x + \frac{1}{3} \cdot \frac{1}{3}) + \frac{2}{3}$

Step 1.3.1.2

Combine $x$ and $\frac{1}{3}$ .

$f (x) = 3 (x^{2} + \frac{x}{3} + \frac{1}{3} x + \frac{1}{3} \cdot \frac{1}{3}) + \frac{2}{3}$

Step 1.3.1.3

Combine $\frac{1}{3}$ and $x$ .

$f (x) = 3 (x^{2} + \frac{x}{3} + \frac{x}{3} + \frac{1}{3} \cdot \frac{1}{3}) + \frac{2}{3}$

Step 1.3.1.4

Multiply $\frac{1}{3} \cdot \frac{1}{3}$ .

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Step 1.3.1.4.1

Multiply $\frac{1}{3}$ by $\frac{1}{3}$ .

$f (x) = 3 (x^{2} + \frac{x}{3} + \frac{x}{3} + \frac{1}{3 \cdot 3}) + \frac{2}{3}$

Step 1.3.1.4.2

Multiply $3$ by $3$ .

$f (x) = 3 (x^{2} + \frac{x}{3} + \frac{x}{3} + \frac{1}{9}) + \frac{2}{3}$

Step 1.3.2

Add $\frac{x}{3}$ and $\frac{x}{3}$ .

$f (x) = 3 (x^{2} + 2 (\frac{x}{3}) + \frac{1}{9}) + \frac{2}{3}$

Step 1.4

Combine $2$ and $\frac{x}{3}$ .

$f (x) = 3 (x^{2} + \frac{2 x}{3} + \frac{1}{9}) + \frac{2}{3}$

Step 1.5

Apply the distributive property.

$f (x) = 3 x^{2} + 3 (\frac{2 x}{3}) + 3 (\frac{1}{9}) + \frac{2}{3}$

Step 1.6

Simplify.

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Step 1.6.1

Cancel the common factor of $3$ .

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Step 1.6.1.1

Cancel the common factor.

$f (x) = 3 x^{2} + 3 (\frac{2 x}{3}) + 3 (\frac{1}{9}) + \frac{2}{3}$

Step 1.6.1.2

Rewrite the expression.

$f (x) = 3 x^{2} + 2 x + 3 (\frac{1}{9}) + \frac{2}{3}$

Step 1.6.2

Cancel the common factor of $3$ .

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Step 1.6.2.1

Factor $3$ out of $9$ .

$f (x) = 3 x^{2} + 2 x + 3 (\frac{1}{3 (3)}) + \frac{2}{3}$

Step 1.6.2.2

Cancel the common factor.

$f (x) = 3 x^{2} + 2 x + 3 (\frac{1}{3 \cdot 3}) + \frac{2}{3}$

Step 1.6.2.3

Rewrite the expression.

$f (x) = 3 x^{2} + 2 x + \frac{1}{3} + \frac{2}{3}$

Step 2

Combine fractions.

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Step 2.1

Combine the numerators over the common denominator.

$f (x) = 3 x^{2} + 2 x + \frac{1 + 2}{3}$

Step 2.2

Simplify the expression.

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Step 2.2.1

Add $1$ and $2$ .

$f (x) = 3 x^{2} + 2 x + \frac{3}{3}$

Step 2.2.2

Divide $3$ by $3$ .

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

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 $b$ .

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

Step 4.2

Remove parentheses.

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

Step 4.3

Cancel the common factor of $2$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 5

Evaluate $f (- \frac{1}{3})$ .

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Step 5.1

Replace the variable $x$ with $- \frac{1}{3}$ in the expression.

$f (- \frac{1}{3}) = 3 {(- \frac{1}{3})}^{2} + 2 (- \frac{1}{3}) + 1$

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1.1.1

Apply the product rule to $- \frac{1}{3}$ .

$f (- \frac{1}{3}) = 3 ({(- 1)}^{2} {(\frac{1}{3})}^{2}) + 2 (- \frac{1}{3}) + 1$

Step 5.2.1.1.2

Apply the product rule to $\frac{1}{3}$ .

$f (- \frac{1}{3}) = 3 ({(- 1)}^{2} (\frac{1^{2}}{3^{2}})) + 2 (- \frac{1}{3}) + 1$

Step 5.2.1.2

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

$f (- \frac{1}{3}) = 3 (1 (\frac{1^{2}}{3^{2}})) + 2 (- \frac{1}{3}) + 1$

Step 5.2.1.3

Multiply $\frac{1^{2}}{3^{2}}$ by $1$ .

$f (- \frac{1}{3}) = 3 (\frac{1^{2}}{3^{2}}) + 2 (- \frac{1}{3}) + 1$

Step 5.2.1.4

One to any power is one.

$f (- \frac{1}{3}) = 3 (\frac{1}{3^{2}}) + 2 (- \frac{1}{3}) + 1$

Step 5.2.1.5

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

$f (- \frac{1}{3}) = 3 (\frac{1}{9}) + 2 (- \frac{1}{3}) + 1$

Step 5.2.1.6

Cancel the common factor of $3$ .

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Step 5.2.1.6.1

Factor $3$ out of $9$ .

$f (- \frac{1}{3}) = 3 (\frac{1}{3 (3)}) + 2 (- \frac{1}{3}) + 1$

Step 5.2.1.6.2

Cancel the common factor.

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

Step 5.2.1.6.3

Rewrite the expression.

$f (- \frac{1}{3}) = \frac{1}{3} + 2 (- \frac{1}{3}) + 1$

Step 5.2.1.7

Multiply $2 (- \frac{1}{3})$ .

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Step 5.2.1.7.1

Multiply $- 1$ by $2$ .

$f (- \frac{1}{3}) = \frac{1}{3} - 2 (\frac{1}{3}) + 1$

Step 5.2.1.7.2

Combine $- 2$ and $\frac{1}{3}$ .

$f (- \frac{1}{3}) = \frac{1}{3} + \frac{- 2}{3} + 1$

Step 5.2.1.8

Move the negative in front of the fraction.

$f (- \frac{1}{3}) = \frac{1}{3} - \frac{2}{3} + 1$

Step 5.2.2

Combine fractions.

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Step 5.2.2.1

Combine the numerators over the common denominator.

$f (- \frac{1}{3}) = 1 + \frac{1 - 2}{3}$

Step 5.2.2.2

Simplify the expression.

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Step 5.2.2.2.1

Subtract $2$ from $1$ .

$f (- \frac{1}{3}) = 1 + \frac{- 1}{3}$

Step 5.2.2.2.2

Move the negative in front of the fraction.

$f (- \frac{1}{3}) = 1 - \frac{1}{3}$

Step 5.2.2.2.3

Write $1$ as a fraction with a common denominator.

$f (- \frac{1}{3}) = \frac{3}{3} - \frac{1}{3}$

Step 5.2.2.2.4

Combine the numerators over the common denominator.

$f (- \frac{1}{3}) = \frac{3 - 1}{3}$

Step 5.2.2.2.5

Subtract $1$ from $3$ .

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

Step 5.2.3

The final answer is $\frac{2}{3}$ .

$\frac{2}{3}$

Step 6

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

$(- \frac{1}{3}, \frac{2}{3})$

Step 7