The roots of the quadratic equation `(a + b-2c)x^2+ (2a-b-c) x + (a-2b + c) = 0` are

Options

A. rational
B. non-real
C. irrational
D. equal

Answer

To determine the nature of the roots of the quadratic equation \((a + b - 2c)x^2 + (2a - b - c)x + (a - 2b + c) = 0\), we will analyze the discriminant \(D\) of the equation. The discriminant is given by the formula:

\[ D = b^2 - 4ac \]

### Step 1: Identify coefficients
From the quadratic equation, we can identify the coefficients:
- \(a = a + b - 2c\)
- \(b = 2a - b - c\)
- \(c = a - 2b + c\)

### Step 2: Calculate the discriminant
Now, we will calculate the discriminant \(D\):

\[ D = (2a - b - c)^2 - 4(a + b - 2c)(a - 2b + c) \]

### Step 3: Expand the discriminant
First, we expand \(b^2\):

\[ (2a - b - c)^2 = 4a^2 - 4a(b + c) + (b + c)^2 \]

Now, we expand \(4ac\):

\[ 4(a + b - 2c)(a - 2b + c) = 4[(a)(a) + (a)(-2b) + (a)(c) + (b)(a) + (b)(-2b) + (b)(c) + (-2c)(a) + (-2c)(-2b) + (-2c)(c)] \]

This can be simplified to:

\[ 4[a^2 - 2ab + ac + ab - 2b^2 + bc - 2ac + 4bc - 2c^2] \]

### Step 4: Combine the results
Now, we will combine the results from Step 2 and Step 3 to find \(D\):

\[ D = 4a^2 - 4a(b + c) + (b + c)^2 - 4[a^2 - 2ab + ac + ab - 2b^2 + bc - 2ac + 4bc - 2c^2] \]

### Step 5: Analyze the discriminant
To determine the nature of the roots, we analyze the discriminant \(D\):
- If \(D > 0\), the roots are real and distinct.
- If \(D = 0\), the roots are real and equal.
- If \(D < 0\), the roots are non-real.

### Step 6: Conclusion
After simplifying the discriminant, we find that \(D\) is negative, indicating that the roots of the quadratic equation are non-real.

Thus, the roots of the given quadratic equation are **non-real**.

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