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explain completing squre of this equation

The image shows the process of completing the square for the quadratic equation \(3x^2 + 4x + 5 = 3\). Let's break down each step:

### Step 1: Start with the original equation
\[
3x^2 + 4x + 5 = 3
\]

### Step 2: Factor out the coefficient of \(x^2\) (which is 3) from the quadratic and linear terms
\[
3 \left( x^2 + \frac{4}{3}x \right) + 5 = 3
\]

Rewrite the equation as:
\[
3 \left( x^2 + \frac{4}{3}x + \frac{5}{3} \right) = 3
\]

### Step 3: Complete the square inside the brackets
To complete the square for \(x^2 + \frac{4}{3}x\), find the term that makes it a perfect square trinomial:
- Take half of the coefficient of \(x\), which is \(\frac{4}{3}\), and square it:
\[
\left( \frac{2}{3} \right)^2 = \frac{4}{9}
\]

Add and subtract \(\frac{4}{9}\) inside the brackets:
\[
3 \left( x^2 + \frac{4}{3}x + \frac{4}{9} - \frac{4}{9} + \frac{5}{3} \right)
\]

### Step 4: Rewrite as a perfect square
Now, the expression inside the brackets can be written as:
\[
3 \left( \left( x + \frac{2}{3} \right)^2 - \frac{4}{9} + \frac{5}{3} \right)
\]

### Step 5: Simplify the constants
Combine \(-\frac{4}{9} + \frac{5}{3}\):
\[
-\frac{4}{9} + \frac{15}{9} = \frac{11}{9}
\]

So, the equation becomes:
\[
3 \left( \left( x + \frac{2}{3} \right)^2 + \frac{11}{9} \right) = 3
\]

### Step 6: Rewrite the final equation
The final expression is:
\[
3 \left( \left( x + \frac{2}{3} \right)^2 + \left( \frac{\sqrt{11}}{3} \right)^2 \right)
\]

This completes the square for the original quadratic equation.

Would you like a more detailed explanation or have any questions?

### Related Questions:
1. How do you complete the square for a quadratic equation without a leading coefficient of 1?
2. What are the applications of completing the square in solving quadratic equations?
3. How does completing the square help in deriving the quadratic formula?
4. Can completing the square be used to graph parabolas? If so, how?
5. What is the geometric interpretation of completing the square?

### Tip:
When completing the square, always remember to add and subtract the same value inside the equation to maintain equality.

explain completing square of this equation

Learn how to complete the square for the quadratic equation 3x^2 + 4x + 5 with a detailed step-by-step explanation. This covers algebraic concepts and solving techniques for students in Grades 8-10.

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