After The School

Deriving and applying the quadratic formula, and understanding the discriminant.

From the method of completing the square, we derive the Quadratic Formula (also known as Shreedharacharya's Rule):

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The expression under the square root, $D = b^2 - 4ac$ , is called the discriminant.

Discriminant	Nature of Roots
$D > 0$	Two distinct real roots
$D = 0$	Two equal real roots (repeated root)
$D < 0$	No real roots

Find the nature of roots of $2x^2 - 4x + 3 = 0$ .

D = b^2 - 4ac = (-4)^2 - 4(2)(3) = 16 - 24 = -8

Since $D < 0$ , the equation has no real roots.

Solve $3x^2 - 5x + 2 = 0$ using the quadratic formula.

x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(2)}}{2(3)} = \frac{5 \pm \sqrt{25 - 24}}{6} = \frac{5 \pm 1}{6}

x = \frac{5 + 1}{6} = 1 \quad \text{or} \quad x = \frac{5 - 1}{6} = \frac{2}{3}

The Quadratic Formula