If quadratic equation `ax^2+bx+c=0` does not have real roots, then which of the following may be false (a) `a(a-b+c) gt 0` (b) `c(a- b+c)>0` e) `b(a- b+c) gt 0` (d) `a +b+c) (a-b+c)gt0`

Options

A. `a(a-b+c)gt 0`
B. `c (a-b +c) gt 0`
C. `b (a-b+c) gt 0`
D. `(a+b+c) (a-b+c) gt 0`

Answer

To determine which of the given statements may be false when the quadratic equation \( ax^2 + bx + c = 0 \) does not have real roots, we need to analyze the implications of the condition that the quadratic has no real roots. This occurs when the discriminant \( D = b^2 - 4ac < 0 \).

Let's analyze each option step by step:

### Step 1: Analyze the Condition for No Real Roots
The condition for the quadratic equation \( ax^2 + bx + c = 0 \) to not have real roots is:
\[ D = b^2 - 4ac < 0 \]
This implies that \( b^2 < 4ac \).

### Step 2: Evaluate Each Option

**Option (a): \( a(a - b + c) > 0 \)**

1. If \( a > 0 \):
- For \( a - b + c > 0 \): This can be true if \( b^2 < 4ac \) holds.
- Thus, \( a(a - b + c) > 0 \) can be true.

2. If \( a < 0 \):
- For \( a - b + c < 0 \): This can also hold true under certain conditions.
- Thus, \( a(a - b + c) > 0 \) can also be true.

**Conclusion**: This option can be true in both cases.

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**Option (b): \( c(a - b + c) > 0 \)**

1. If \( c > 0 \):
- For \( a - b + c > 0 \): This can hold under certain conditions.
- Thus, \( c(a - b + c) > 0 \) can be true.

2. If \( c < 0 \):
- For \( a - b + c < 0 \): This can also hold true under certain conditions.
- Thus, \( c(a - b + c) > 0 \) can also be true.

**Conclusion**: This option can be true in both cases.

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**Option (c): \( b(a - b + c) > 0 \)**

1. If \( b > 0 \):
- For \( a - b + c > 0 \): This can hold true under certain conditions.
- Thus, \( b(a - b + c) > 0 \) can be true.

2. If \( b < 0 \):
- For \( a - b + c < 0 \): This can also hold true under certain conditions.
- Thus, \( b(a - b + c) > 0 \) can also be true.

3. However, since \( b \) can take any value (positive, negative, or zero), we cannot definitively conclude that this expression will always be positive or negative.

**Conclusion**: This option may be false depending on the value of \( b \).

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**Option (d): \( (a + b + c)(a - b + c) > 0 \)**

1. If both \( a + b + c > 0 \) and \( a - b + c > 0 \): This can hold true under certain conditions.
2. If both \( a + b + c < 0 \) and \( a - b + c < 0 \): This can also hold true under certain conditions.

**Conclusion**: This option can be true in both cases.

### Final Conclusion
The option that may be false is:
**(c) \( b(a - b + c) > 0 \)**