We are tasked with completing the square for the quadratic expression $2x^2 - 20x + m$ and using that to match it with the form $p(x - q)^2 + 3q$, which will allow us to find the value of $m$.

Step-by-Step Solution:

1. Factor out the leading coefficient (2): The expression is $2x^2 - 20x + m$. We factor out the 2 from the terms involving $x$: $2(x^2 - 10x) + m$

2. Complete the square inside the parentheses: To complete the square, we look at the $x^2 - 10x$ term. Take half of the coefficient of $x$ (which is $-10$), square it, and add and subtract it inside the parentheses: $x^2 - 10x = (x - 5)^2 - 25$ So the expression becomes: $2((x - 5)^2 - 25) + m = 2(x - 5)^2 - 50 + m$

3. Rearrange the expression: Now we rewrite it as: $2(x - 5)^2 + (m - 50)$

4. Compare this with the given form: The given form is $p(x - q)^2 + 3q$. From this, we see that: $p = 2, \quad q = 5, \quad \text{and} \quad 3q = m - 50$ Substituting $q = 5$ into $3q$, we get: $3(5) = m - 50 \quad \Rightarrow \quad 15 = m - 50$

5. Solve for $m$: Add 50 to both sides: $m = 65$

Thus, the value of $m$ is 65.

Would you like more details or clarifications on any step?

Related Questions:

1. How do you complete the square for a quadratic expression?
2. Why do we factor out the leading coefficient before completing the square?
3. What is the general formula for completing the square?
4. How does completing the square help in solving quadratic equations?
5. What are other methods to solve quadratic expressions, apart from completing the square?

Tip:

Completing the square is also useful in finding the vertex of a quadratic function, as the expression $a(x-h)^2 + k$ gives the vertex $(h, k)$.