For `a, b, c in R` and a, b, c are different, then the roots of `(x+a)(x+b)= c^(2)` are

Options

A. real and equal
B. real and unequal
C. complex number
D. Cannot be determined

Answer

To solve the equation \((x + a)(x + b) = c^2\) for the roots, we will follow these steps:

### Step 1: Expand the Left Side
We start by expanding the left side of the equation:
\[ (x + a)(x + b) = x^2 + (a + b)x + ab \]
Thus, the equation becomes:
\[ x^2 + (a + b)x + ab = c^2 \]

### Step 2: Rearrange the Equation
Next, we rearrange the equation to set it to zero:
\[ x^2 + (a + b)x + (ab - c^2) = 0 \]

### Step 3: Identify Coefficients
From the standard form of a quadratic equation \(Ax^2 + Bx + C = 0\), we identify:
- \(A = 1\)
- \(B = a + b\)
- \(C = ab - c^2\)

### Step 4: Calculate the Discriminant
The discriminant \(D\) of a quadratic equation is given by:
\[ D = B^2 - 4AC \]
Substituting the values we identified:
\[ D = (a + b)^2 - 4 \cdot 1 \cdot (ab - c^2) \]
This simplifies to:
\[ D = (a + b)^2 - 4ab + 4c^2 \]

### Step 5: Simplify the Discriminant
Now, we simplify the discriminant:
\[ D = a^2 + 2ab + b^2 - 4ab + 4c^2 \]
\[ D = a^2 - 2ab + b^2 + 4c^2 \]
This can be factored as:
\[ D = (a - b)^2 + 4c^2 \]

### Step 6: Analyze the Discriminant
Since \(a\), \(b\), and \(c\) are all different real numbers, we know:
- \((a - b)^2 \geq 0\) (it is always non-negative)
- \(4c^2 > 0\) (since \(c \neq 0\))

Thus, the discriminant \(D\) is always greater than zero:
\[ D > 0 \]

### Step 7: Conclusion About the Roots
Since the discriminant is positive, we conclude that the roots of the quadratic equation are:
- Real and unequal.

### Final Answer
Therefore, the roots of the equation \((x + a)(x + b) = c^2\) are real and unequal.

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