Quadratic Equations

James Blackburn

January 2022

1 Proving the α β formulas

Define the quadratic equation with the variables a,b,c ax2 + bx+ c = 0 Let α = x1 (The first r√oot of the equation) --b+--b2 --4ac α = 2a Let β = x2 (The second root of the equation) √-2----- β = - b---b---4ac 2a Prove the formula α +β = --b is true (via substitution) √------- a √------- --b+--b2-- 4ac + --b--b2 --4ac 2a 2a - 2b- = 2a - b = a Prove the formula αβ = c-is true (via substitution) √ ------a √------- --b+---b2---4ac --b---b2 --4ac 2a × 2a (- b + √b2 --4ac)(- b- √b2---4ac) = -------------4a2------------- 2 √ -2-----2 = (--b)--- (-b2---4ac) 24a 2 = b---(b---4ac)- 4a2 = 4ac 4a2 = c- a
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