Question

$\sf{\red{Brahmaputra\: Quadratic\: Equations}}$
$\pink{\boxed{\sf{{\huge\mathscr{\red x}} \: = \frac{\pm \sqrt{4\blue{a}{c} + \green{b^{2}}} \: -\green{ b}}{2\blue{a}}}}}$
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Answer 1

$\implies\sf\red{Case I: b – 4ac > 0}$

When a, b and c are real numbers, a ≠ 0 and discriminant is positive (i.e., b2 – 4ac > 0), then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real and unequal.

$\implies\sf\orange{Case II: b2 – 4ac = 0}$

When a, b and c are real numbers, a ≠ 0 and discriminant is zero (i.e., b2 – 4ac = 0), then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real and equal.

$\implies\sf\green{Case III: b2 – 4ac < 0}$

When a, b and c are real numbers, a ≠ 0 and discriminant is negative (i.e., b2 – 4ac < 0), then the roots α and β of the quadratic equation ax2 + bx + c = 0 are unequal and imaginary. Here the roots α and β are a pair of the complex conjugates.

$\rightarrow\sf\blue{Case IV: b2 – 4ac > 0 and~ perfect ~square}$

When a, b and c are real numbers, a ≠ 0 and discriminant is positive and perfect square, then the roots α and β of the quadratic equation ax2 + bx + c = 0 are real, rational unequal.

$\sf\pink{Case V: b2 – 4ac > 0 and ~not~ perfect ~square}$

When a, b and c are real numbers, a ≠ 0 and discriminant is positive but not a perfect square then the roots of the quadratic equation ax2 + bx + c = 0 are real, irrational and unequal.

Here the roots α and β form a pair of irrational conjugates.