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c) 2x² + 5x - 12 = 0
a = 2, b=5, c = -12
Δ= b² - 4ac = 5² -4*2 *(-12)= 25 + 96 = 121
√Δ= 11
-b-√Δ -5 -11
x1 = ----------- = ---------------- = -16/4 = -4
2a 2 * 2
x2= (-b + √Δ ) / 2a = (-5 + 11) / 2*2 = 6/4 = 1 2/4 = 1 1/2
x1= - 4, x2 = 1 1/2
d) 3x² + 7x - 20 = 0
a = 3, b = 7, c = -20
Δ= b² - 4ac = 7² - 4 * 3 * (-20) = 49 + 240 = 289
√Δ= 17
x1= (-b-√Δ) /2a = (-7 -17) / 2 * 3 = -24 /6 = -4
x2= (-b+√Δ) / 2a = ( -7 + 17) / 2*3 = 10/6 = 1 4/6 = 1 2/3
zad.103
a)
3x² + 5x = 2
3x² + 5x - 2 = 0
a = 3, b = 5, c = -2
Δ = b² - 4ac = 5² -4 * 3 * (-2) = 25 + 24 = 49
√Δ = 7
x1= (-5 -7) / 2* 3 = -12 /6 = -2
x2 = (-5 + 7) / 2* 3 = 2 /6 = 1/3
b) 5x - 3 = 2x²
-2x² + 5x - 3 = 0
a = -2, b = 5 c = -3
Δ= b³ - 4ac = 5² - 4 * (-2) * (-3) = 25 - 24 = 1
x1= (-5 -1) / 2 *(-2) = -6 / -4 = 1 2/4 = 1 1/2
x2 = (-5 + 1) / 2 *(-2) = -4 / -4 = 1
c) 2x² + 3x + 8 = 0
a = 2, b = 3, c = 8
Δ = b² - 4ac = 3² - 4 * 2 * 8 = 9 - 64 = -55
Δ = - 55
rownanie nie ma rozwiazania gdyz Δ < 0
d) 5x² - 2 - 9x
5x² - 9x - 2 = 0
a = 5, b = -9, c = -2
Δ = b² - 4ac = (-9)² - 4 * 5 * (-2) = 81 + 40 = 121
√Δ = 11
x1 = [-(-9 ) -11] / 2 * 5 = (9 - 11) / 10 = -2 /10 = -1/5
x2 = 9 + 11 / 2*5 = 20 / 10 = 2
e) - x² + 3x = 5
- x² + 3x - 5 = 0
a = -1, b = 3, c = - 5
Δ = b² -4ac = 3² - 4 * (-1) * (-5) = 9 - 20 = -11
Δ = -11 to Δ < 0
brak rozwiazania delta mniejsza od zera
f) 6x² = x + 1
6x² - x - 1 = 0
a = 6, b = -1, c = -1
Δ= b² - 4ac = (-1)² - 4 * 6 * (-1) = 1 + 24 = 25
√Δ = 5
x1 = - (-1) - 5 / 2 *6 = 1 - 5 / 12 = -4 /12 = - 1/3
x2 = 1 + 5 / 2* 6 = 6 / 12 = 1/2
zad.104
b)
1/2x² - 1,3x - 3/5 = 0
1/2x² - 13/10x - 3/5 = 0 / *10
5x² - 13x - 6 = 0
a = 5, b = -13, c = - 6
Δ = b² - 4ac = (-13)² - 4 * 5 * (-6) = 169 + 120 = 289
√Δ = 17
x1 = - (-13) - 17 / 2 * 5 = 13-17 / 10 = -4 /10 = - 2/5
x2 = 13 + 17 / 2*5 = 30 / 10 = 3
c)
5,5x - 1 - 4,5x² = 0
- 4,5x² + 5,5x - 1 = 0
a = -4,5, b= 5,5, c = -1
Δ= b² - 4 ac = 5,5² - 4 * (-4,5) * (-1) = 30,25 - 18 = 12,25
√Δ = 3,5
x1 = -5,5 - 3,5 / 2 * (-4,5) = - 9 / -9 = 1
x2 = -5,5 + 3,5 / 2 * (-4,5) = -2 / -9 = 2/9
d)
-1,5x² + 3,5 x = -3
-1,5x² + 3,5x + 3 = 0
a = -1,5 , b = 3,5 , c = 3
Δ= b² - 4 ac = 3,5² - 4 * (-1,5) * 3 = 12,25 + 18 = 30,25
√Δ = 5,5
x1= (-b -√Δ) / 2a = (-3,5 - 5,5) / 2 * (-1,5) = -9 / -3 = 3
x2 =(-b +√Δ ) / 2a = -3,5 + 5,5 / 2* (-1,5) = 2 / -3 = -2/3