Express y = − 32/(20)^2x^2 + x + 6 in standard form.

To express the given equation 

\[
y = -\frac{32}{20^2}x^2 + x + 6
\]

in standard form, we need to simplify and rewrite the equation as a quadratic expression in the form:

\[
y = ax^2 + bx + c
\]

### Step 1: Simplify the fraction

First, simplify the fraction \(\frac{32}{20^2}\).

\[
20^2 = 400, \quad \text{so} \quad \frac{32}{20^2} = \frac{32}{400}
\]

Simplify \(\frac{32}{400}\):

\[
\frac{32}{400} = \frac{32 \div 16}{400 \div 16} = \frac{2}{25}
\]

So, the equation simplifies to:

\[
y = -\frac{2}{25}x^2 + x + 6
\]

### Step 2: Write in standard form

The simplified equation is already in the standard form \(ax^2 + bx + c\):

\[
y = -\frac{2}{25}x^2 + 1x + 6
\]

where:
- \(a = -\frac{2}{25}\)
- \(b = 1\)
- \(c = 6\)

The standard form of the equation is:

\[
\boxed{y = -\frac{2}{25}x^2 + x + 6}
\]

Would you like to go into more detail, or do you have any other questions? Here are some related questions you might be interested in:

1. How do you find the vertex of a quadratic equation in standard form?
2. How can you determine the axis of symmetry of this quadratic function?
3. What is the process to convert a quadratic equation from standard form to vertex form?
4. How do you solve for the roots of the quadratic equation given above?
5. What is the effect of changing the coefficient \(a\) on the graph of the parabola?

**Tip:** To check the behavior of a quadratic equation, it is helpful to understand how each term affects the graph (i.e., stretching, shifting, and direction).

Simplify and express the quadratic equation y = − 32/(20)^2x^2 + x + 6 in standard form. This problem involves algebraic concepts such as simplifying fractions and writing quadratic equations in standard form. Suitable for students in Grades 8-10.

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