The value of 'a' for which the quadratic equation `2x^(2) - x(a^(2) + 8a -1) + a^(2) - 4a = 0` has roots with opposite signs, lie in the interval

Options

A. `1 lt a lt 5`
B. `0 lt a lt 4`
C. `-1 lt a lt 2`
D. `2 lt a lt 6`

Answer

To solve the quadratic equation \(2x^2 - x(a^2 + 8a - 1) + (a^2 - 4a) = 0\) for the value of \(a\) such that the roots have opposite signs, we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic equation can be written in the standard form \(Ax^2 + Bx + C = 0\):
- \(A = 2\)
- \(B = -(a^2 + 8a - 1)\)
- \(C = a^2 - 4a\)

### Step 2: Condition for opposite signs of roots
For the roots of a quadratic equation to have opposite signs, the product of the roots must be negative. The product of the roots for a quadratic equation is given by \(\frac{C}{A}\). Thus, we need:
\[ \frac{C}{A} < 0 \]
Substituting the values of \(C\) and \(A\):
\[ \frac{a^2 - 4a}{2} < 0 \]

### Step 3: Simplify the inequality
Multiplying both sides of the inequality by 2 (which does not change the inequality since 2 is positive):
\[ a^2 - 4a < 0 \]

### Step 4: Factor the quadratic expression
Factoring the left-hand side:
\[ a(a - 4) < 0 \]

### Step 5: Find the critical points
The critical points of the inequality \(a(a - 4) = 0\) are:
- \(a = 0\)
- \(a = 4\)

### Step 6: Test intervals
We will test the intervals defined by the critical points \(0\) and \(4\):
1. For \(a < 0\) (e.g., \(a = -1\)):
- \(a(-1 - 4) = -1(-5) > 0\) (not valid)
2. For \(0 < a < 4\) (e.g., \(a = 2\)):
- \(2(2 - 4) = 2(-2) < 0\) (valid)
3. For \(a > 4\) (e.g., \(a = 5\)):
- \(5(5 - 4) = 5(1) > 0\) (not valid)

### Step 7: Conclusion
The solution to the inequality \(a(a - 4) < 0\) is:
\[ 0 < a < 4 \]
Thus, the values of \(a\) for which the quadratic equation has roots with opposite signs lie in the interval \((0, 4)\).

### Final Answer
The value of \(a\) for which the quadratic equation has roots with opposite signs lies in the interval \( (0, 4) \).

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