Write the equation in standard form to find the center and radius of the circle. Then sketch the graph-
x2+y2+4x=21

To write the equation of the circle in standard form and find its center and radius, we follow these steps:

### Given equation:
\[
x^2 + y^2 + 4x = 21
\]

### Step 1: Rearrange the equation
Move all terms involving \(x\) and \(y\) on the left-hand side and constants on the right-hand side:
\[
x^2 + 4x + y^2 = 21
\]

### Step 2: Complete the square for the \(x\)-terms
We need to complete the square for the \(x\)-terms in the equation:
- The \(x\)-terms are \(x^2 + 4x\).
- Take half of the coefficient of \(x\), which is 4, divide by 2 to get 2, and square it to get \(4\).
- Add and subtract 4 inside the equation:
\[
x^2 + 4x + 4 + y^2 = 21 + 4
\]

### Step 3: Simplify
Now simplify the equation:
\[
(x + 2)^2 + y^2 = 25
\]

### Step 4: Identify the center and radius
The equation is now in standard form:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
where \((h, k)\) is the center and \(r\) is the radius.

From the equation \((x + 2)^2 + y^2 = 25\):
- The center is \((-2, 0)\).
- The radius is \(r = \sqrt{25} = 5\).

### Final Answer:
The center of the circle is \((-2, 0)\), and the radius is 5.

### Sketch of the graph:
- The circle is centered at \((-2, 0)\).
- The radius is 5, so the circle extends 5 units from the center in all directions.

Let me know if you'd like me to generate a visual sketch of this graph.

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Would you like further clarification or details on any part of the process?

Here are 5 related questions to explore further:

1. How do you complete the square in quadratic expressions?
2. How do you convert general form to standard form for a circle equation?
3. How do you find the radius and center from a given equation of a circle?
4. What is the significance of the standard form of a circle's equation in graphing?
5. How do you graph a circle given its center and radius?

**Tip:** Completing the square is a crucial technique in algebra for transforming quadratic equations into a more manageable form.

Write the equation in standard form to find the center and radius of the circle. Then sketch the graph- x2 + y2 + 4x = 21

Learn how to convert the equation of a circle x^2 + y^2 + 4x = 21 to its standard form. This problem covers essential geometry concepts, including completing the square to find the center (-2, 0) and radius (5). Suitable for students in Grades 9-11.

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