Convert the following general form to standard form equation of an hyperbola 8x²-5y²+16x-32= 0

We are tasked with converting the general form equation of a hyperbola:

\[
8x^2 - 5y^2 + 16x - 32 = 0
\]

to its standard form. Follow these steps:

### Step 1: Group the \(x\) and \(y\) terms.
We start by grouping the \(x\)-terms and \(y\)-terms:

\[
(8x^2 + 16x) - 5y^2 = 32
\]

### Step 2: Factor out the coefficient of \(x^2\).
We factor out \(8\) from the \(x\)-terms:

\[
8(x^2 + 2x) - 5y^2 = 32
\]

### Step 3: Complete the square for the \(x\)-terms.
To complete the square for \(x^2 + 2x\), we take half of \(2\), which is \(1\), and square it to get \(1\). So, we add and subtract \(1\) inside the parentheses:

\[
8(x^2 + 2x + 1 - 1) - 5y^2 = 32
\]

Simplifying:

\[
8((x + 1)^2 - 1) - 5y^2 = 32
\]

### Step 4: Distribute the constants.
Now, distribute the \(8\):

\[
8(x + 1)^2 - 8 - 5y^2 = 32
\]

Simplify further:

\[
8(x + 1)^2 - 5y^2 = 40
\]

### Step 5: Divide by the constant on the right-hand side.
To convert this into standard form, we divide everything by \(40\):

\[
\frac{8(x + 1)^2}{40} - \frac{5y^2}{40} = 1
\]

Simplifying the fractions:

\[
\frac{(x + 1)^2}{5} - \frac{y^2}{8} = 1
\]

### Final Standard Form:

\[
\frac{(x + 1)^2}{5} - \frac{y^2}{8} = 1
\]

This is the standard form of the given hyperbola.

Would you like more details on any specific step, or do you have any questions?

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Here are 5 related questions:
1. How do you determine the center of a hyperbola from its standard form?
2. What are the key differences between the equations of ellipses and hyperbolas?
3. How do you calculate the foci of a hyperbola given its standard form?
4. Can the orientation of a hyperbola change based on its equation? How?
5. How is the asymptote of a hyperbola derived from its standard form?

**Tip**: For hyperbolas, the terms involving squares have opposite signs (one positive, one negative), which distinguishes them from ellipses.

Convert the following general form to standard form equation of a hyperbola 8x²-5y²+16x-32= 0

Convert the general form equation 8x²-5y²+16x-32= 0 into its standard form step-by-step. This problem explores hyperbolas, conic sections, and completing the square, essential concepts for high school students in Grades 10-12.

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