Find the Maximum/Minimum Value y=(x-5)^2-7

$y = {(x - 5)}^{2} - 7$

Step 1

Simplify each term.

Rewrite ${(x - 5)}^{2}$ as $(x - 5) (x - 5)$ .

$f (x) = (x - 5) (x - 5) - 7$

Expand $(x - 5) (x - 5)$ using the FOIL Method.

Apply the distributive property.

$f (x) = x (x - 5) - 5 (x - 5) - 7$

Apply the distributive property.

$f (x) = x \cdot x + x \cdot - 5 - 5 (x - 5) - 7$

Apply the distributive property.

$f (x) = x \cdot x + x \cdot - 5 - 5 x - 5 \cdot - 5 - 7$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = x^{2} + x \cdot - 5 - 5 x - 5 \cdot - 5 - 7$

Move $- 5$ to the left of $x$ .

$f (x) = x^{2} - 5 \cdot x - 5 x - 5 \cdot - 5 - 7$

Multiply $- 5$ by $- 5$ .

$f (x) = x^{2} - 5 x - 5 x + 25 - 7$

Subtract $5 x$ from $- 5 x$ .

$f (x) = x^{2} - 10 x + 25 - 7$

Step 2

Subtract $7$ from $25$ .

$f (x) = x^{2} - 10 x + 18$

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 $x = - \frac{b}{2 a}$ .

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

$x = - \frac{- 10}{2 (1)}$

Remove parentheses.

$x = - \frac{- 10}{2 (1)}$

Simplify $- \frac{- 10}{2 (1)}$ .

Cancel the common factor of $- 10$ and $2$ .

Factor $2$ out of $- 10$ .

$x = - \frac{2 \cdot - 5}{2 \cdot 1}$

Cancel the common factors.

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

$x = - \frac{2 \cdot - 5}{2 (1)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 5}{2 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 5}{1}$

Divide $- 5$ by $1$ .

$x = - - 5$

Multiply $- 1$ by $- 5$ .

$x = 5$

Step 5

Evaluate $f (5)$ .

Replace the variable $x$ with $5$ in the expression.

$f (5) = {(5)}^{2} - 10 \cdot 5 + 18$

Simplify the result.

Simplify each term.

Raise $5$ to the power of $2$ .

$f (5) = 25 - 10 \cdot 5 + 18$

Multiply $- 10$ by $5$ .

$f (5) = 25 - 50 + 18$

Simplify by adding and subtracting.

Subtract $50$ from $25$ .

$f (5) = - 25 + 18$

Add $- 25$ and $18$ .

$f (5) = - 7$

The final answer is $- 7$ .

$- 7$

Step 6

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

$(5, - 7)$

Step 7

Find the Maximum/Minimum Value y=(x-5)^2-7

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