The values of m for which the equation `5x^(2)-4x+2+m(4x^(2)-2x-1)=0` will have (i) Equal roots (ii) Product of roots as 2 (iii) Sum of the roots as 6 are ……. And ……..

Answer

To solve the given problem, we need to analyze the quadratic equation:

\[ 5x^2 - 4x + 2 + m(4x^2 - 2x - 1) = 0 \]

We will rewrite this equation in the standard form \( ax^2 + bx + c = 0 \).

### Step 1: Combine Like Terms
First, we will combine the coefficients of \( x^2 \), \( x \), and the constant term.

- Coefficient of \( x^2 \): \( 5 + 4m \)
- Coefficient of \( x \): \( -4 - 2m \)
- Constant term: \( 2 - m \)

So, the quadratic equation can be rewritten as:

\[ (5 + 4m)x^2 + (-4 - 2m)x + (2 - m) = 0 \]

### Step 2: Equal Roots Condition
For the quadratic equation to have equal roots, the discriminant must be zero. The discriminant \( D \) is given by:

\[ D = b^2 - 4ac \]

Substituting \( a = 5 + 4m \), \( b = -4 - 2m \), and \( c = 2 - m \):

\[ D = (-4 - 2m)^2 - 4(5 + 4m)(2 - m) \]

Expanding this:

\[ D = (16 + 16m + 4m^2) - 4[(10 - 5m + 8m - 4m^2)] \]

\[ D = 16 + 16m + 4m^2 - 4(10 + 3m - 4m^2) \]

\[ D = 16 + 16m + 4m^2 - 40 - 12m + 16m^2 \]

Combining like terms:

\[ D = 20m^2 + 4m - 24 \]

Setting the discriminant equal to zero for equal roots:

\[ 20m^2 + 4m - 24 = 0 \]

### Step 3: Solve the Quadratic Equation
Dividing through by 4:

\[ 5m^2 + m - 6 = 0 \]

Now we can factor this:

\[ (5m - 6)(m + 1) = 0 \]

Setting each factor to zero gives us:

\[ m = \frac{6}{5} \quad \text{or} \quad m = -1 \]

### Step 4: Product of Roots Condition
For the product of the roots to be 2, we use the formula:

\[ \text{Product of roots} = \frac{c}{a} = 2 \]

Substituting \( c = 2 - m \) and \( a = 5 + 4m \):

\[ \frac{2 - m}{5 + 4m} = 2 \]

Cross-multiplying:

\[ 2 - m = 2(5 + 4m) \]

Expanding:

\[ 2 - m = 10 + 8m \]

Rearranging gives:

\[ 2 - 10 = 9m \quad \Rightarrow \quad -8 = 9m \quad \Rightarrow \quad m = -\frac{8}{9} \]

### Step 5: Sum of Roots Condition
For the sum of the roots to be 6, we use the formula:

\[ \text{Sum of roots} = -\frac{b}{a} = 6 \]

Substituting \( b = -4 - 2m \) and \( a = 5 + 4m \):

\[ -\frac{-4 - 2m}{5 + 4m} = 6 \]

This simplifies to:

\[ \frac{4 + 2m}{5 + 4m} = 6 \]

Cross-multiplying:

\[ 4 + 2m = 6(5 + 4m) \]

Expanding:

\[ 4 + 2m = 30 + 24m \]

Rearranging gives:

\[ 4 - 30 = 22m \quad \Rightarrow \quad -26 = 22m \quad \Rightarrow \quad m = -\frac{13}{11} \]

### Final Answers
The values of \( m \) for the three conditions are:

1. Equal roots: \( m = \frac{6}{5} \) or \( m = -1 \)
2. Product of roots = 2: \( m = -\frac{8}{9} \)
3. Sum of roots = 6: \( m = -\frac{13}{11} \)