The roots of the equation `x^(2)-2ax+8a-15=0` are equal, then a =

Options

A. 3 or 2
B. 3 or 5
C. 5 or 2
D. `+-2`

Answer

To find the values of \( a \) for which the roots of the equation \( x^2 - 2ax + 8a - 15 = 0 \) are equal, we can follow these steps:

### Step 1: Identify coefficients
The given quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here, we can identify:
- \( a = 1 \)
- \( b = -2a \)
- \( c = 8a - 15 \)

### Step 2: Use the condition for equal roots
For the roots of a quadratic equation to be equal, the discriminant must be zero. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Setting the discriminant equal to zero gives:
\[ (-2a)^2 - 4(1)(8a - 15) = 0 \]

### Step 3: Simplify the equation
Calculating the discriminant:
\[ 4a^2 - 4(8a - 15) = 0 \]
Distributing the \( -4 \):
\[ 4a^2 - 32a + 60 = 0 \]

### Step 4: Factor out the common term
We can factor out \( 4 \):
\[ 4(a^2 - 8a + 15) = 0 \]
Dividing both sides by \( 4 \):
\[ a^2 - 8a + 15 = 0 \]

### Step 5: Factor the quadratic
Now we can factor the quadratic:
\[ (a - 5)(a - 3) = 0 \]

### Step 6: Solve for \( a \)
Setting each factor to zero gives us:
\[ a - 5 = 0 \quad \Rightarrow \quad a = 5 \]
\[ a - 3 = 0 \quad \Rightarrow \quad a = 3 \]

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

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