If the equation `mx^(2)+mx+1=-4x^(2)-x` has equal roots, then the values of m are

Options

A. `-5,3`
B. `5,-3`
C. `5,3`
D. `-5,-3`

Answer

To solve the equation \( mx^2 + mx + 1 = -4x^2 - x \) for the values of \( m \) when the equation has equal roots, we can follow these steps:

### Step 1: Rearranging the Equation
First, we need to rearrange the given equation into standard quadratic form \( ax^2 + bx + c = 0 \).

Starting with:
\[ mx^2 + mx + 1 + 4x^2 + x = 0 \]
Combine like terms:
\[ (m + 4)x^2 + (m + 1)x + 1 = 0 \]

### Step 2: Identifying Coefficients
In the standard form \( ax^2 + bx + c = 0 \), we identify:
- \( a = m + 4 \)
- \( b = m + 1 \)
- \( c = 1 \)

### Step 3: Condition for Equal Roots
For the quadratic equation to have equal roots, the discriminant \( D \) must be zero. The discriminant is given by:
\[ D = b^2 - 4ac \]
Substituting the values of \( a \), \( b \), and \( c \):
\[ D = (m + 1)^2 - 4(m + 4)(1) \]

### Step 4: Expanding the Discriminant
Now, we expand the discriminant:
\[ D = (m + 1)^2 - 4(m + 4) \]
Calculating \( (m + 1)^2 \):
\[ D = m^2 + 2m + 1 - 4m - 16 \]
Combining like terms:
\[ D = m^2 - 2m - 15 \]

### Step 5: Setting the Discriminant to Zero
For equal roots, set the discriminant \( D \) to zero:
\[ m^2 - 2m - 15 = 0 \]

### Step 6: Factoring the Quadratic Equation
Next, we factor the quadratic equation:
\[ (m - 5)(m + 3) = 0 \]

### Step 7: Finding the Values of \( m \)
Setting each factor to zero gives us the possible values of \( m \):
1. \( m - 5 = 0 \) → \( m = 5 \)
2. \( m + 3 = 0 \) → \( m = -3 \)

Thus, the values of \( m \) for which the equation has equal roots are:
\[ m = 5 \quad \text{or} \quad m = -3 \]