Let's solve the problem step by step:

Step 1: Rewrite the equation of the circle

The given equation is:

$5x^2 + 10x + 5y^2 = 0$

Divide the entire equation by $5$ to simplify:

$x^2 + 2x + y^2 = 0$

This is the simplified form of the equation.

Step 2: Complete the square to find the center and radius

To complete the square, focus on the $x$-terms ($x^2 + 2x$):

1. Add and subtract $\left(\frac{2}{2}\right)^2 = 1$ to $x^2 + 2x$: $x^2 + 2x = (x + 1)^2 - 1$

2. Rewrite the equation with this completed square form: $(x + 1)^2 - 1 + y^2 = 0$

3. Simplify further: $(x + 1)^2 + y^2 = 1$

Step 3: Identify the center and radius

The equation $(x + 1)^2 + y^2 = 1$ represents a circle with:

• Center $(h, k) = (-1, 0)$
• Radius $r = \sqrt{1} = 1$

Step 4: Graph the circle

The circle has:

• A center at $(-1, 0)$
• A radius of $1$, so it will pass through points $(-2, 0)$, $(0, 0)$, $(-1, 1)$, and $(-1, -1)$.

Step 5: Find the intercepts

1. x-intercepts: Set $y = 0$ in $(x + 1)^2 + y^2 = 1$: $(x + 1)^2 = 1 \implies x + 1 = \pm 1$ Solve for $x$: $x = 0 \quad \text{and} \quad x = -2$ So the x-intercepts are $(0, 0)$ and $(-2, 0)$.

2. y-intercepts: Set $x = 0$ in $(x + 1)^2 + y^2 = 1$: $(0 + 1)^2 + y^2 = 1 \implies 1 + y^2 = 1$ $y^2 = 0 \implies y = 0$ So the y-intercept is $(0, 0)$.

Final Results:

1. Center: $(-1, 0)$
2. Radius: $1$
3. x-intercepts: $(0, 0)$, $(-2, 0)$
4. y-intercept: $(0, 0)$

Do you want me to:

• Graph the circle for you?
• Provide further details on how intercepts were calculated?

Related Questions:

1. How do you find the standard equation of a circle from general form?
2. What are the steps to complete the square in a quadratic equation?
3. How do intercepts of a circle relate to its radius?
4. Can a circle's equation ever have no intercepts? Why or why not?
5. How would this equation change if it represented an ellipse instead of a circle?

Tip: Always simplify equations first to make the process of finding intercepts and centers easier!