x²-(P+3)X+12=0
has roots of a and b, a=3b. Find the value of p

We are given the quadratic equation:

\[
x^2 - (P + 3)x + 12 = 0
\]

with roots \( a \) and \( b \), and it is also given that \( a = 3b \).

We can use Vieta's formulas, which relate the coefficients of a quadratic equation to its roots:

1. The sum of the roots \( a + b = P + 3 \)
2. The product of the roots \( a \cdot b = 12 \)

### Step 1: Use the relationship between \( a \) and \( b \)
Since \( a = 3b \), we substitute this into the formulas from Vieta's relations.

#### Sum of the roots:
\[
a + b = 3b + b = 4b
\]
So,
\[
4b = P + 3
\]

#### Product of the roots:
\[
a \cdot b = 3b \cdot b = 3b^2
\]
We are given that the product of the roots is 12, so:
\[
3b^2 = 12
\]
\[
b^2 = 4
\]
\[
b = 2 \quad \text{(since the roots are usually positive in such cases)}
\]

### Step 2: Solve for \( a \) and \( P \)
Now that we have \( b = 2 \), we can find \( a \):
\[
a = 3b = 3 \times 2 = 6
\]

Now substitute \( b = 2 \) into the equation for the sum of the roots:
\[
4b = P + 3
\]
\[
4 \times 2 = P + 3
\]
\[
8 = P + 3
\]
\[
P = 5
\]

Thus, the value of \( P \) is \( \boxed{5} \).

Would you like a more detailed explanation on any part of the process?

---

Here are 5 related questions for further practice:

1. Solve the quadratic equation \( x^2 - 7x + 10 = 0 \) using Vieta's relations.
2. If \( x^2 - (k + 4)x + 15 = 0 \) has roots in the ratio 1:2, find \( k \).
3. Find the roots of the quadratic equation \( x^2 - 9x + 20 = 0 \) using factoring.
4. How would the value of \( P \) change if the relationship between \( a \) and \( b \) were \( a = 2b \) instead?
5. Derive Vieta's formulas for the general quadratic equation \( ax^2 + bx + c = 0 \).

**Tip:** When working with roots of quadratic equations, always check if Vieta's relations apply, as they offer a quicker way to connect coefficients with roots.

The query submitted by the user: x² - (P + 3)x + 12 = 0 has roots of a and b, a = 3b. Find the value of p.

Discover how to solve the quadratic equation x² - (P + 3)x + 12 = 0 where the roots a and b are related by a = 3b. This problem explores key algebraic techniques such as Vieta's formulas and quadratic equations, making it suitable for students in Grades 8-10.

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