If incentre of triangle whose vertices are $(0, 0), (4, 0), (0, -3)$ is ($\alpha, \beta$), then $\alpha, \beta$ are roots of the equation .

The given vertices (0, 0), (4, 0), and (0, negative 3) form a right-angled triangle with the right angle at the origin. The lengths of the sides forming the right angle are 4 and 3, making the hypotenuse 5 by the Pythagorean theorem. The coordinates of the incenter (alpha, beta) of a triangle are given by the weighted average of its vertices using the side lengths as weights, so alpha equals (5 multiplied by 0 plus 3 multiplied by 4 plus 4 multiplied by 0) divided by the perimeter 12, which is 1. Similarly, beta equals (5 multiplied by 0 plus 3 multiplied by 0 plus 4 multiplied by negative 3) divided by 12, which equals negative 1. Since the incentre coordinates are (1, negative 1), alpha and beta are exactly 1 and negative 1. These values are the roots of the quadratic equation whose sum is 0 and product is negative 1, giving x squared minus 1 equals 0.

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