If α & β are roots of quadratic equation x2 – x + (m + 1) = 0 & roots are in the ratio 3 : 2, then value of 10(m + 1) is

Let the quadratic equation be \( x^2 - x + (m + 1) = 0 \).

The roots of this equation are given as \( \alpha \) and \( \beta \), and they are in the ratio 3:2. This means:

\[
\alpha = 3k \quad \text{and} \quad \beta = 2k
\]

Now, from Vieta's formulas for a quadratic equation \( ax^2 + bx + c = 0 \), we know:

- The sum of the roots \( \alpha + \beta = -\frac{b}{a} \)
- The product of the roots \( \alpha \cdot \beta = \frac{c}{a} \)

For our equation \( x^2 - x + (m + 1) = 0 \), the sum and product of the roots will be:

1. Sum of the roots \( \alpha + \beta = 1 \) (since the coefficient of \( x \) is -1)
2. Product of the roots \( \alpha \cdot \beta = m + 1 \) (since the constant term is \( m + 1 \))

### Step 1: Find \( k \)
From the sum of the roots:
\[
\alpha + \beta = 3k + 2k = 5k = 1
\]
Thus:
\[
k = \frac{1}{5}
\]

### Step 2: Find \( m \)
Now, for the product of the roots:
\[
\alpha \cdot \beta = (3k) \cdot (2k) = 6k^2
\]
Using the value of \( k = \frac{1}{5} \):
\[
\alpha \cdot \beta = 6 \left( \frac{1}{5} \right)^2 = 6 \times \frac{1}{25} = \frac{6}{25}
\]

We are also given that:
\[
\alpha \cdot \beta = m + 1
\]
So:
\[
m + 1 = \frac{6}{25}
\]
Thus:
\[
m = \frac{6}{25} - 1 = \frac{6}{25} - \frac{25}{25} = \frac{-19}{25}
\]

### Step 3: Calculate \( 10(m + 1) \)
We are asked to find \( 10(m + 1) \). Since \( m + 1 = \frac{6}{25} \):
\[
10(m + 1) = 10 \times \frac{6}{25} = \frac{60}{25} = 2.4
\]

Thus, the value of \( 10(m + 1) \) is \( \boxed{2.4} \).

Do you have any questions or need further details? Here are some related problems for practice:

1. Solve a quadratic equation where the roots are in the ratio 4:3.
2. If the sum of the roots of a quadratic equation is 5 and the product is 6, find the quadratic equation.
3. Derive the quadratic equation from given roots.
4. Solve for \( m \) if the roots are related by a different ratio.
5. Explore the geometric significance of roots' ratios in quadratic equations.

**Tip:** Always double-check Vieta’s relations; they offer a powerful tool for connecting roots with coefficients.

Learn how to solve the quadratic equation x^2 - x + (m + 1) = 0 where the roots are in the ratio 3:2. Follow a detailed step-by-step solution using Vieta's formulas to find the value of 10(m + 1). Suitable for students in Grades 9-12.

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