Squares And Quadratics

Since the roots ae integers , so discriminant = $p^2-4q$ is a perfect square.

For mathematical experiment for $x^2+px+q=0$ . By using the completing the square we can determine the perfect square. So the perfect square for $x^2+px+q=0$ is $p^2-4q$ that is a discriminant.

$\LARGE \text{ADIOS!!!}$

For the quadratic equation $ax^2 + bx + c = 0$ to contain integer roots, its determinant $\sqrt{b^2 - 4ac}$ must be a perfect square. In the quadratic equation above where p and q are integers, we can substitute its values for a, b, and c into the discriminant.

For $x^2 + px + q$ , the discriminant can be expressed as: $\sqrt{p^2 - 4q}$ . Thus, $p^2 - 4q$ must be a perfect square.

-p = a + b, p^2 = a^2 + 2ab + b^2, q = ab, 4q = 4ab, p^2 - 4q = a^2 + 2ab + b^2 - 4ab, p^2 - 4q = a^2 - 2ab + b^2