We are given the equation:

$x^2 + 8x + 10 = (x + p)^2 + q$

We need to find the values of $p$ and $q$. Let's break this down.

Step 1: Expand the right-hand side

We know that:

$(x + p)^2 = x^2 + 2px + p^2$

Thus, the equation becomes:

$x^2 + 8x + 10 = x^2 + 2px + p^2 + q$

Step 2: Compare coefficients

By comparing the two sides of the equation, we match terms.

• The coefficient of $x^2$ on both sides is 1, so it cancels out.

• The coefficient of $x$ gives us: 8 = 2p \quad \text{(so, p = 4)}.

• The constant terms give us: 10 = p^2 + q \quad \text{(since p = 4, substitute into this)}. $10 = 4^2 + q$ $10 = 16 + q$ $q = 10 - 16 = -6$

Final values

• $p = 4$
• $q = -6$

Would you like further details on this explanation? Here are 5 questions related to this concept:

1. How do you complete the square in a quadratic equation?
2. What is the relationship between completing the square and finding the vertex of a parabola?
3. Can you solve for $p$ and $q$ if the given equation involved different coefficients?
4. What are the applications of completing the square in solving quadratic equations?
5. How does completing the square help in understanding the geometric properties of a quadratic function?

Tip: When solving quadratics, it's often helpful to complete the square if you're looking to transform the equation into a perfect square form, particularly for vertex form or solving.