Let `a ,b ,c` be distinct non-negative numbers and the vectors `a hat i+a hat j+c hat k , hat i+ hat k ,c hat i+c hat j+b hat k` lie in a plane, and then prove that the quadratic equation `a x^2+2c x+b=0` has equal roots.

Explanation

To prove that the quadratic equation \( ax^2 + 2cx + b = 0 \) has equal roots, we need to show that its discriminant is zero. We start by using the condition that the vectors given lie in the same plane, which implies they are coplanar.

### Step 1: Set up the vectors
The vectors given are:
1. \( \mathbf{v_1} = a \hat{i} + a \hat{j} + c \hat{k} \)
2. \( \mathbf{v_2} = \hat{i} + \hat{k} \)
3. \( \mathbf{v_3} = c \hat{i} + c \hat{j} + b \hat{k} \)

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