Question

-2x^2 + 5x + 12

Answer 1

It looks like you have a quadratic expression: \(-2x^2 + 5x + 12\). If you're looking to analyze or manipulate this expression in some way, here are a few things you might be interested in:

1. **Factoring**: This expression can be challenging to factor directly because the leading coefficient is negative. We could use the quadratic formula to find its roots.

2. **Finding Roots**: You can find the roots of the equation \(-2x^2 + 5x + 12 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -2\), \(b = 5\), and \(c = 12\):

\[
x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot (-2) \cdot 12}}{2 \cdot (-2)}
\]

\[
= \frac{-5 \pm \sqrt{25 + 96}}{-4}
\]

\[
= \frac{-5 \pm \sqrt{121}}{-4}
\]

\[
= \frac{-5 \pm 11}{-4}
\]

This gives two solutions:

- \(x = \frac{6}{-4} = -\frac{3}{2}\)
- \(x = \frac{-16}{-4} = 4\)

3. **Vertex**: The vertex of a quadratic function in the form \(ax^2 + bx + c\) can be found using the formula \(x = -\frac{b}{2a}\):

\[
x = -\frac{5}{2 \cdot -2} = \frac{5}{4}
\]

You can find the \(y\) coordinate of the vertex by substituting \(x = \frac{5}{4}\) back into the quadratic expression.

4. **Graphing**: You can graph the quadratic function, noting that it opens downwards (since the leading coefficient -2 is negative).

If you have a specific question or task in mind regarding this quadratic expression, please let me know!