Algebra Examples

$g (x) = 2 x (x - 4) + 7$

Step 1

Simplify each term.

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Step 1.1

Apply the distributive property.

$g (x) = 2 x \cdot x + 2 x \cdot - 4 + 7$

Step 1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.2.1

Move $x$ .

$g (x) = 2 (x \cdot x) + 2 x \cdot - 4 + 7$

Step 1.2.2

Multiply $x$ by $x$ .

$g (x) = 2 x^{2} + 2 x \cdot - 4 + 7$

Step 1.3

Multiply $- 4$ by $2$ .

$g (x) = 2 x^{2} - 8 x + 7$

Step 2

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 3

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

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Step 3.1

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

$x = - \frac{- 8}{2 (2)}$

Step 3.2

Remove parentheses.

$x = - \frac{- 8}{2 (2)}$

Step 3.3

Simplify $- \frac{- 8}{2 (2)}$ .

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Step 3.3.1

Cancel the common factor of $- 8$ and $2$ .

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Step 3.3.1.1

Factor $2$ out of $- 8$ .

$x = - \frac{2 \cdot - 4}{2 \cdot 2}$

Step 3.3.1.2

Cancel the common factors.

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Step 3.3.1.2.1

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

$x = - \frac{2 \cdot - 4}{2 (2)}$

Step 3.3.1.2.2

Cancel the common factor.

$x = - \frac{2 \cdot - 4}{2 \cdot 2}$

Step 3.3.1.2.3

Rewrite the expression.

$x = - \frac{- 4}{2}$

Step 3.3.2

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

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Step 3.3.2.1

Factor $2$ out of $- 4$ .

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

Step 3.3.2.2

Cancel the common factors.

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Step 3.3.2.2.1

Factor $2$ out of $2$ .

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

Step 3.3.2.2.2

Cancel the common factor.

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

Step 3.3.2.2.3

Rewrite the expression.

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

Step 3.3.2.2.4

Divide $- 2$ by $1$ .

$x = - - 2$

Step 3.3.3

Multiply $- 1$ by $- 2$ .

$x = 2$

Step 4

Evaluate $f (2)$ .

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Step 4.1

Replace the variable $x$ with $2$ in the expression.

$g (2) = 2 {(2)}^{2} - 8 \cdot 2 + 7$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

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Step 4.2.1.1.1

Multiply $2$ by ${(2)}^{2}$ .

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Step 4.2.1.1.1.1

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

$g (2) = 2 {(2)}^{2} - 8 \cdot 2 + 7$

Step 4.2.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$g (2) = 2^{1 + 2} - 8 \cdot 2 + 7$

Step 4.2.1.1.2

Add $1$ and $2$ .

$g (2) = 2^{3} - 8 \cdot 2 + 7$

Step 4.2.1.2

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

$g (2) = 8 - 8 \cdot 2 + 7$

Step 4.2.1.3

Multiply $- 8$ by $2$ .

$g (2) = 8 - 16 + 7$

Step 4.2.2

Simplify by adding and subtracting.

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Step 4.2.2.1

Subtract $16$ from $8$ .

$g (2) = - 8 + 7$

Step 4.2.2.2

Add $- 8$ and $7$ .

$g (2) = - 1$

Step 4.2.3

The final answer is $- 1$ .

$- 1$

Step 5

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

$(2, - 1)$

Step 6