Question

Derive quadratic formula.

Answer 1

let the quadratic be
ax²+bx+c=0
x²+(b/a)x+c/a = 0
x² +2×(b/2a)x +c/a= 0
x² +2×(b/2a)x + (b/2a)² - (b/2a)² + c/a= 0
(x+(b/2a))² - b²/4a² +c/a = 0
(x+(b/2a))² = b²/4a² -c/a
(x+(b/2a))² = (b²-4ac)/4a²
take square root,
x + (b/2a) = ±√(b²-4ac)/2a

x = -(b/2a)±√(b²-4ac)/2a

Hence, proved!

Answer 2

General form of Quadratic equation is
ax² + bx + c = 0

derivation of formulae for finding roots of Quadratic equation.
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ax² + bx + c = 0

ax² + bx = -c

divide both sides by a

x² + (b/a).x = -c/a

add both sides, ( b/2a)²

x² + 2.(b/2a).x + (b/2a)² = -c + (b/2a)²

[ use, formula , a² + 2ab + b² = (a + b)² ]

( x + b/2a)² = ( b² - 4ac)/4a²

take square root both sides,

(x + b/2a) = ± √(b² -4ac)/2a

x = -b/2a ± √(b² - 4ac)/2a

x = { - b ±√(b² - 4ac)}/2a