f ( x )

To rewrite the quadratic function $f(x) = -2x^2 - 5x - 6$ in standard form (also called vertex form), we need to complete the square. The vertex form of a quadratic function is:

$f(x) = a(x-h)^2 + k$

where $(h, k)$ is the vertex of the parabola. Let's go through the steps to complete the square.

Step 1: Factor out the coefficient of $x^2$ from the first two terms

Since the coefficient of $x^2$ is $-2$, we factor it out:

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

Step 2: Complete the square

To complete the square, take half of the coefficient of $x$ (which is $\frac{5}{2}$), square it, and add it inside the parentheses. Half of $\frac{5}{2}$ is $\frac{5}{4}$, and squaring it gives $\frac{25}{16}$.

We add and subtract $\frac{25}{16}$ inside the parentheses:

$f(x) = -2\left(x^2 + \frac{5}{2}x + \frac{25}{16} - \frac{25}{16}\right) - 6$

Step 3: Simplify the expression inside the parentheses

The expression inside the parentheses can be factored as a perfect square:

$f(x) = -2\left(\left(x + \frac{5}{4}\right)^2 - \frac{25}{16}\right) - 6$

Step 4: Distribute the $-2$ and simplify

Now distribute the $-2$ across both terms inside the parentheses:

$f(x) = -2\left(x + \frac{5}{4}\right)^2 + 2 \times \frac{25}{16} - 6$

Simplify the constant terms:

$f(x) = -2\left(x + \frac{5}{4}\right)^2 + \frac{50}{16} - 6$

Convert $-6$ to a fraction with a denominator of 16:

$f(x) = -2\left(x + \frac{5}{4}\right)^2 + \frac{50}{16} - \frac{96}{16}$

Now simplify the constant term:

$f(x) = -2\left(x + \frac{5}{4}\right)^2 - \frac{46}{16}$

Finally, simplify the constant:

$f(x) = -2\left(x + \frac{5}{4}\right)^2 - \frac{23}{8}$

Final Answer:

The quadratic function in vertex form is:

$f(x) = -2\left(x + \frac{5}{4}\right)^2 - \frac{23}{8}$

Would you like more details or clarification on any part? Here are some related questions:

1. How can you find the vertex of a quadratic function directly?
2. What does the vertex form tell you about the graph of the quadratic?
3. How can you derive the quadratic formula from completing the square?
4. What does the sign of the coefficient of $x^2$ indicate about the parabola?
5. Can all quadratic functions be rewritten in vertex form?

Tip: Completing the square is a powerful tool not just for rewriting functions but also for solving quadratic equations.