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The question from your image appears to be **Question 55**, which is related to solving and verifying roots of a quadratic equation. The main parts seem to be:

### Part (a):
You are asked to find the values of \( p \) and \( q \), where the equation is given in factored form. It appears the equation is of the form:

\[
(x - p)(x - q) = 0
\]

The roots of the quadratic are \( p \) and \( q \), and you need to deduce their values from the expanded form or factored form.

### Part (b):
Harry claims that the roots of the equation \( x^2 + 5x + 3 = 0 \) are \( p = -1.5 \) and \( q = -2.5 \). You are supposed to check this claim.

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Let's break this down.

#### For part (a):

1. You seem to have factored the quadratic equation and found:
   \[
   p = -1.5 \quad \text{and} \quad q = -2.5
   \]
   You can verify these values by multiplying the factors back to the quadratic form:
   \[
   (x + 1.5)(x + 2.5) = x^2 + 4x + 3.75
   \]
   However, Harry seems to claim that the equation is \( x^2 + 5x + 3 = 0 \), so this doesn't perfectly match his equation.

#### For part (b):

You should check if Harry's roots satisfy the equation by substituting them into the quadratic formula or solving using the discriminant method. If you expand Harry's factorization and it matches the quadratic equation, then his claim is valid.

Would you like to go into the specific details of each part to verify the answers?

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### Expansion Questions:
1. How do we determine the discriminant of a quadratic equation, and what does it tell us about the nature of the roots?
2. What would be the effect of slight variations in the coefficient values on the roots of a quadratic equation?
3. Can we always factor a quadratic equation, or are there cases where we must use the quadratic formula instead?
4. How do you verify if roots are real or complex?
5. What is the importance of ensuring the equation is in standard form before attempting to factor it?

### Tip:
Always double-check the expanded form after factoring or solving, to ensure your roots satisfy the original quadratic equation!

55. It is given that p and q are the roots of the quadratic equation x^2 + 5x + 3 = 0. (a) Find the values of p and q. (b) Harry claims that the roots are p = -1.5 and q = -2.5. Do you agree? Explain.

Understand how to find the roots of the quadratic equation x^2 + 5x + 3 = 0 and verify the correctness of given roots using factorization and the quadratic formula. Includes a step-by-step solution suitable for students in Grades 9-10.

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