∆ = b² − 4ac → se numeste discriminant al ecuatiei de gradul al doilea.
a) x² - 7x - 18 = 0
Δ=(-7)²- 4×1×(-18) =49+72 = 121
x₁= [ -(-7)+√121] / 2 = (7 + 11) / 2 =18/2 =9
x₂ = (7-11)/2= - 4 /2 = -2
b) 3x² - 5x + 2 = 0
Δ= (-5)² -4×3×2=25-24=1
x₁= [ -(-5) + √1] / 2×3= (5+1) / 6 =6/6=1
x₂= ( 5 -1)/6 = 4/6 = 2/3=0,(6)
c) 3x² - 11x + 10 = 0
Δ= (-11)² - 4×3×10=121-120=1
x₁=[- (-11)+√1] / 2×3 = 12/6=2
x₂= [ - ( -11) -√1)/6=10/6=5/3=1,(6)
d) 4x² - 4x + 1 = 0
Δ= (-4)² - 4×4×1=16-16=0
Cum Δ=0 ⇒ x₁=x₂= - (-4) / 2×4=4/8=1/2=0,5
e) x² + 3x + 5 = 0
Δ=3²-4×1×5=9-20=-11
Cum Δ<0, ecuatia nu are solutii reale S=Ф
f) 2x - x² + 3 = 0 ⇔ -x²+2x+3=0
Δ= 2²- 4 ×(-1)×3=4 + 12 =16
x₁= ( -2 +√16) / 2×(-1) = (-2+4) / -2 = 6/ - 2 = -3
x₂= (-2 -4) / -2 = -6/-2=3
g) 1 - x - 6x² = 0⇔ -6x²-x +1=0
Δ=(-1)² - 4 × 1× (-6) = 1 +24 =25
x₁= [ - (-1) +√25] / 2×(-6)= (1+5)/-12 = -1/2= - 0,5
h) 25x² + 10x +1 = 0
Δ=10² - 4×25×1= 100 -100=0
Cum Δ=0 ⇒ x₁=x₂= - 10/ 2×25 = - 1/5
i) x² + 7x - 1 = 0
Δ=7²-4×1×(-1)=49 +4=53
x₁= (-7+√53)/ 2=( - 7 + 7,28) /2= - 0,14
x₂ = ( - 7 - 7,28)/2= - 14,28 /2 = - 7,14
