If a, b, c are real then `(b-x)^(2)-4(a-x) (c-x)=0` will have always roots which are

Options

A. real and equal
B. real and distinct
C. imaginary
D. both 1 and 2

Answer

To determine the nature of the roots of the equation \((b-x)^{2} - 4(a-x)(c-x) = 0\), we will follow these steps:

### Step 1: Expand the Equation
Start by expanding the left-hand side of the equation.

\[ (b - x)^{2} - 4(a - x)(c - x) = 0 \]

Expanding \((b - x)^{2}\):
\[ (b - x)^{2} = b^{2} - 2bx + x^{2} \]

Expanding \(4(a - x)(c - x)\):
\[ 4(a - x)(c - x) = 4(ac - ax - cx + x^{2}) = 4ac - 4ax - 4cx + 4x^{2} \]

Putting it all together:
\[ b^{2} - 2bx + x^{2} - (4ac - 4ax - 4cx + 4x^{2}) = 0 \]

### Step 2: Combine Like Terms
Now, combine the terms:
\[ b^{2} - 2bx + x^{2} - 4ac + 4ax + 4cx - 4x^{2} = 0 \]

This simplifies to:
\[ b^{2} - 4ac + (4a - 2b + 4c)x - 3x^{2} = 0 \]

### Step 3: Rearrange into Standard Quadratic Form
Rearranging gives us:
\[ -3x^{2} + (4a - 2b + 4c)x + (b^{2} - 4ac) = 0 \]

This is a quadratic equation of the form \(Ax^{2} + Bx + C = 0\), where:
- \(A = -3\)
- \(B = 4a - 2b + 4c\)
- \(C = b^{2} - 4ac\)

### Step 4: Find the Discriminant
To determine the nature of the roots, we calculate the discriminant \(D\):
\[ D = B^{2} - 4AC \]

Substituting the values:
\[ D = (4a - 2b + 4c)^{2} - 4(-3)(b^{2} - 4ac) \]

### Step 5: Simplify the Discriminant
Calculating \(D\):
\[ D = (4a - 2b + 4c)^{2} + 12(b^{2} - 4ac) \]

### Step 6: Analyze the Discriminant
The expression \((4a - 2b + 4c)^{2}\) is always non-negative since it is a square. The term \(12(b^{2} - 4ac)\) can be positive, negative, or zero depending on the values of \(a\), \(b\), and \(c\).

However, the term \(b^{2} - 4ac\) is a standard form that can be analyzed further.

### Conclusion
Since \((4a - 2b + 4c)^{2}\) is always non-negative and \(12(b^{2} - 4ac)\) can also yield a positive discriminant, we conclude that the discriminant \(D\) is always greater than or equal to zero.

Thus, the roots of the equation are **real and distinct** when \(D > 0\) and **real and equal** when \(D = 0\).

### Final Answer
The roots of the equation will always be **real and distinct**.

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