Solve v^2-16v-48=-4 by completing square 

To solve the quadratic equation \( v^2 - 16v - 48 = -4 \) by completing the square, follow these steps:

### Step 1: Move the constant term to the other side
Start by adding 48 to both sides of the equation to get rid of the constant term on the left-hand side:

\[
v^2 - 16v = 44
\]

### Step 2: Complete the square
To complete the square, take half of the coefficient of \( v \), which is \( -16 \), and then square it:

\[
\left( \frac{-16}{2} \right)^2 = (-8)^2 = 64
\]

Add this value to both sides of the equation:

\[
v^2 - 16v + 64 = 44 + 64
\]

Simplifying:

\[
v^2 - 16v + 64 = 108
\]

### Step 3: Express the left-hand side as a perfect square
The left-hand side is now a perfect square trinomial:

\[
(v - 8)^2 = 108
\]

### Step 4: Solve for \( v \)
Take the square root of both sides:

\[
v - 8 = \pm \sqrt{108}
\]

Since \( \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} \), we have:

\[
v - 8 = \pm 6\sqrt{3}
\]

### Step 5: Solve for \( v \)
Now, solve for \( v \) by adding 8 to both sides:

\[
v = 8 \pm 6\sqrt{3}
\]

Thus, the solutions are:

\[
v = 8 + 6\sqrt{3} \quad \text{or} \quad v = 8 - 6\sqrt{3}
\]

### Final Answer:
\[
v = 8 \pm 6\sqrt{3}
\]

Would you like more details or clarification on any step?

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Here are 5 related questions to expand your understanding:
1. How do you convert a quadratic equation into vertex form?
2. Can completing the square always be used to solve quadratic equations?
3. What is the geometric interpretation of completing the square?
4. How do you solve a quadratic equation by using the quadratic formula instead?
5. What is the connection between completing the square and the formula for the vertex of a parabola?

**Tip:** Completing the square is a useful technique not just for solving equations, but also for rewriting functions to identify their vertex.

Solve v^2 - 16v - 48 = -4 by completing the square

Learn how to solve the quadratic equation v^2 - 16v - 48 = -4 by completing the square in this step-by-step guide. This solution involves key algebraic techniques such as completing the square, transforming the equation into a perfect square trinomial, and using the square root method to find the roots. Suitable for students in Grades 9-11.

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