Find the Vertex -3(x+5)^2-1.5

$- 3 {(x + 5)}^{2} - 1.5$

Step 1

Rewrite the equation in vertex form.

Complete the square for $- 3 {(x + 5)}^{2} - 1.5$ .

Simplify the expression.

Simplify each term.

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

$- 3 ((x + 5) (x + 5)) - 1.5$

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

Apply the distributive property.

$- 3 (x (x + 5) + 5 (x + 5)) - 1.5$

Apply the distributive property.

$- 3 (x \cdot x + x \cdot 5 + 5 (x + 5)) - 1.5$

Apply the distributive property.

$- 3 (x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5) - 1.5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$- 3 (x^{2} + x \cdot 5 + 5 x + 5 \cdot 5) - 1.5$

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

$- 3 (x^{2} + 5 \cdot x + 5 x + 5 \cdot 5) - 1.5$

Multiply $5$ by $5$ .

$- 3 (x^{2} + 5 x + 5 x + 25) - 1.5$

Add $5 x$ and $5 x$ .

$- 3 (x^{2} + 10 x + 25) - 1.5$

Apply the distributive property.

$- 3 x^{2} - 3 (10 x) - 3 \cdot 25 - 1.5$

Simplify.

Multiply $10$ by $- 3$ .

$- 3 x^{2} - 30 x - 3 \cdot 25 - 1.5$

Multiply $- 3$ by $25$ .

$- 3 x^{2} - 30 x - 75 - 1.5$

Subtract $1.5$ from $- 75$ .

$- 3 x^{2} - 30 x - 76.5$

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

$a = - 3$

$b = - 30$

$c = - 76.5$

Consider the vertex form of a parabola.

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

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

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

$d = \frac{- 30}{2 \cdot - 3}$

Simplify the right side.

Cancel the common factor of $- 30$ and $2$ .

Factor $2$ out of $- 30$ .

$d = \frac{2 \cdot - 15}{2 \cdot - 3}$

Cancel the common factors.

Factor $2$ out of $2 \cdot - 3$ .

$d = \frac{2 \cdot - 15}{2 (- 3)}$

Cancel the common factor.

$d = \frac{2 \cdot - 15}{2 \cdot - 3}$

Rewrite the expression.

$d = \frac{- 15}{- 3}$

Cancel the common factor of $- 15$ and $- 3$ .

Factor $- 3$ out of $- 15$ .

$d = \frac{- 3 (5)}{- 3}$

Cancel the common factors.

Factor $- 3$ out of $- 3$ .

$d = \frac{- 3 \cdot 5}{- 3 \cdot 1}$

Cancel the common factor.

$d = \frac{- 3 \cdot 5}{- 3 \cdot 1}$

Rewrite the expression.

$d = \frac{5}{1}$

Divide $5$ by $1$ .

$d = 5$

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

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

$e = - 76.5 - \frac{{(- 30)}^{2}}{4 \cdot - 3}$

Simplify the right side.

Simplify each term.

Raise $- 30$ to the power of $2$ .

$e = - 76.5 - \frac{900}{4 \cdot - 3}$

Multiply $4$ by $- 3$ .

$e = - 76.5 - \frac{900}{- 12}$

Divide $900$ by $- 12$ .

$e = - 76.5 - - 75$

Multiply $- 1$ by $- 75$ .

$e = - 76.5 + 75$

Add $- 76.5$ and $75$ .

$e = - 1.5$

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- 3 {(x + 5)}^{2} - 1.5$ .

$- 3 {(x + 5)}^{2} - 1.5$

Set $y$ equal to the new right side.

$y = - 3 {(x + 5)}^{2} - 1.5$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 3$

$h = - 5$

$k = - 1.5$

Step 3

Find the vertex $(h, k)$ .

$(- 5, - 1.5)$

Step 4

Find the Vertex -3(x+5)^2-1.5

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