What is direct evidence

Direct evidence - explanation and example

In the following we want to prove that under certain conditions the $ pq $ formula is the solution of a quadratic equation in normal form.

If we compare the quadratic equation in normal form $ x ^ 2 + px + q = 0 $ with the first binomial formula $ a ^ 2 + 2ab + b ^ 2 = (a + b) ^ 2 $, we can set up the following equation :

$ x ^ 2 + px + \ left (\ frac p2 \ right) ^ 2 = \ left (x + \ frac p2 \ right) ^ 2 $.

We now want to rearrange this equation so that its left side coincides with the left side of the quadratic equation in normal form, i.e. with $ x ^ 2 + px + q $. To do this, we first perform a zero addition and convert the resulting equation as follows:

$ \ begin {array} {lllll} x ^ 2 + px + \ left (\ frac p2 \ right) ^ 2 + qq & = & \ left (x + \ frac p2 \ right) ^ 2 && \ vert - \ left (\ frac p2 \ right) ^ 2 \ x ^ 2 + px + qq & = & \ left (x + \ frac p2 \ right) ^ 2- \ left (\ frac p2 \ right) ^ 2 && \ vert + q \ x ^ 2 + px + q & = & \ left (x + \ frac p2 \ right) ^ 2- \ left (\ frac p2 \ right) ^ 2 + q \ end {array} $

From the equation to be proved, we know that $ x ^ 2 + px + q = 0 $. So we can now set the right side of the derived relationship equal to zero and solve for $ x $. Then follows:

$ \ begin {array} {lllll} \ left (x + \ frac p2 \ right) ^ 2- \ left (\ frac p2 \ right) ^ 2 + q & = & 0 && \ vert + \ left (\ frac p2 \ right) ^ 2 \ \ left (x + \ frac p2 \ right) ^ 2 + q & = & \ left (\ frac p2 \ right) ^ 2 && \ vert -q \ \ left (x + \ frac p2 \ right ) ^ 2 & = & \ left (\ frac p2 \ right) ^ 2-q && \ \ end {array} $

Under the condition that $ \ left (\ frac p2 \ right) ^ 2-q \ geq 0 $ applies, the root is now taken on both sides of the equation:

$ \ begin {array} {lllll} x_ {1 | 2} + \ frac p2 & = & \ pm \ sqrt {\ left (\ frac p2 \ right) ^ 2-q} && \ vert - \ frac p2 \ x_ {1 | 2} & = & - \ frac p2 \ pm \ sqrt {\ left (\ frac p2 \ right) ^ 2-q} && \ end {array} $

To trick the showman, Brahmagupta can give him two values ​​for $ p $ and $ q $ that do not meet the condition $ \ left (\ frac p2 \ right) ^ 2-q \ geq 0 $.