If `alpha` is a real root of the quadratic equation `a x^2+b x+c=0a n dbeta` ils a real root of `- a x^2+b x+c=0,` then show that there is a root `gamma` of equation `(a//2)x^2+b x+c=0` whilch lies between `aa n dbetadot`

Explanation

To solve the problem, we need to show that if \(\alpha\) is a real root of the quadratic equation \( ax^2 + bx + c = 0 \) and \(\beta\) is a real root of the equation \( -ax^2 + bx + c = 0 \), then there exists a root \(\gamma\) of the equation \( \frac{a}{2}x^2 + bx + c = 0 \) that lies between \(\alpha\) and \(\beta\).

### Step-by-Step Solution:

1. **Define the Functions**:
Let \( f(x) = \frac{a}{2}x^2 + bx + c \).

2. **Evaluate \( f(\alpha) \)**:
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