Consider the quadratic: p(x)=-2x^2-4x+6 a) Find the vertex by completing the square b) Find the zeroes/roots of the quadratic analytically (on paper, not on calculator) c) Graph the quadratic d) What are the domain and range of P(x)? e) Where is P(x) increasing? f) Where is P(x) decreasing?

Let's go through each part of the problem one by one:

### Given quadratic function:
\[
p(x) = -2x^2 - 4x + 6
\]

### Part (a): Find the vertex by completing the square

To complete the square, we first factor out \(-2\) from the terms involving \(x\):

\[
p(x) = -2(x^2 + 2x) + 6
\]

Now, to complete the square for \(x^2 + 2x\), we take half of the coefficient of \(x\) (which is 2), square it, and add it inside the parentheses:

\[
\text{Half of 2 is } 1, \text{ and } 1^2 = 1.
\]

So, we rewrite \(x^2 + 2x\) as:

\[
x^2 + 2x = (x+1)^2 - 1
\]

Now substitute this back into the equation:

\[
p(x) = -2((x+1)^2 - 1) + 6
\]

Distribute the \(-2\):

\[
p(x) = -2(x+1)^2 + 2 + 6
\]

Simplify:

\[
p(x) = -2(x+1)^2 + 8
\]

So, the quadratic is now in vertex form:
\[
p(x) = -2(x+1)^2 + 8
\]

The vertex form of a quadratic is \(p(x) = a(x-h)^2 + k\), where \((h, k)\) is the vertex. Here, the vertex is \((-1, 8)\).

### Part (b): Find the zeroes/roots analytically

To find the roots, we solve the quadratic equation \(p(x) = 0\):

\[
-2x^2 - 4x + 6 = 0
\]

Divide through by \(-2\) to simplify:

\[
x^2 + 2x - 3 = 0
\]

Now, we factor the quadratic:

\[
(x+3)(x-1) = 0
\]

Set each factor equal to zero:

\[
x + 3 = 0 \quad \text{or} \quad x - 1 = 0
\]

Thus, the solutions are:

\[
x = -3 \quad \text{and} \quad x = 1
\]

The roots of the quadratic are \(x = -3\) and \(x = 1\).

### Part (c): Graph the quadratic

The graph of \(p(x) = -2(x+1)^2 + 8\) is a parabola that opens downwards (since the coefficient of \(x^2

Solve the quadratic equation -2x^2-4x+6 step-by-step, including how to find the vertex, roots, and graph the parabola. Understand the domain, range, and intervals of increase and decrease. Ideal for students in Grades 9-10.

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