If the roots of the equation `a x^2+2b x+c=0` and `-2sqrt(a c x)+b=0` are simultaneously real, then prove that `b^2=a c`

Explanation

To prove that \( b^2 = ac \) given that the roots of the equations \( ax^2 + 2bx + c = 0 \) and \( -2\sqrt{ac}x + b = 0 \) are simultaneously real, we will analyze both equations step by step.

### Step 1: Analyze the first equation
The first equation is:
\[ ax^2 + 2bx + c = 0 \]
To find the conditions for the roots to be real, we will use the discriminant. The discriminant \( D \) of a quadratic equation \( Ax^2 + Bx + C = 0 \) is given by:
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