Find the valve of 'a' for which the following equations have equal roots.
`x^(2)+(a+3)x+a+6=0`

Answer

To find the value of 'a' for which the equation \( x^2 + (a + 3)x + (a + 6) = 0 \) has equal roots, we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be compared with the standard form \( ax^2 + bx + c = 0 \). Here, we have:
- \( a = 1 \)
- \( b = a + 3 \)
- \( c = a + 6 \)

### Step 2: Use the condition for equal roots
For a quadratic equation to have equal roots, the discriminant must be equal to zero. The discriminant \( D \) is given by the formula:
\[ D = b^2 - 4ac \]
Setting the discriminant to zero:
\[ b^2 - 4ac = 0 \]

### Step 3: Substitute the coefficients into the discriminant
Substituting the values of \( a \), \( b \), and \( c \):
\[ (a + 3)^2 - 4 \cdot 1 \cdot (a + 6) = 0 \]

### Step 4: Expand and simplify the equation
Expanding the equation:
\[ (a + 3)^2 - 4(a + 6) = 0 \]
\[ a^2 + 6a + 9 - 4a - 24 = 0 \]
Combining like terms:
\[ a^2 + 2a - 15 = 0 \]

### Step 5: Factor the quadratic equation
Next, we will factor the quadratic equation \( a^2 + 2a - 15 = 0 \):
\[ (a + 5)(a - 3) = 0 \]

### Step 6: Solve for 'a'
Setting each factor to zero gives us:
1. \( a + 5 = 0 \) → \( a = -5 \)
2. \( a - 3 = 0 \) → \( a = 3 \)

### Conclusion
The values of \( a \) for which the equation has equal roots are:
\[ a = -5 \quad \text{or} \quad a = 3 \]

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