Угол φ между двумя прямыми, заданными уравнениями c угловыми коэффициентами
y=k₁x+b₁ y=k₂x+b₂, вычисляется по формуле: tgφ=(k₂-k₁)/(1+k₁*k₂)
а) y=-3х/4-1 и y=3х/4 +2
tgφ=(3/4+3/4)/(1-9/16)=3*16/(2*7)=24/7=3 3/7
б) 2y+3x-1=0 и 3y+2x-5=0; у=-3х/2 -1/2и у=2х/3 +5/3;
tgφ=(2/3+3/2)/(1-(3*2)*(2/3)); tgφ=∞; φ=90°
в) x = 1 и y = -2x + 1;
cosφ=(1*2+0*1)/(√1*√5)=2/√5; sinφ=√(1-4/5)=1/√5; tgφ=(1/√5):(2/√5)=1/2
г) x = -3 и 3x + 2y - 3 = 0
cosφ=(1*3+0*2)/(√1*√(3²+2²))=3/√13; sinφ=√(1-9/13)=2/√13;
tgφ=(2/√13):(3/√13)=2/3
4x² - 12x + 9 = 0
D = b² - 4ac = 144 - 16×9 = 0
x = -b/2a
x = 12/8
x = 1,5
2) 5x² + 1 - 6x + 4x² = 0
9x² - 6x + 1 = 0
D = b² - 4ac = 36 - 36×1 = 0
x = -b/2a
x = 6/18
x = 1/3
3) x² + 2x - 3 = 0
D = b² -4ac = 4 - 4×(-3) = 26 = 4²
x1 = ( - 2 + 4) / 2 = 1
x2 = ( - 2 - 4) / 2 = - 3
4) x² + 3x -4 = 0
D = b²- 4ac = 9 - 4×(-4) = 25 = 5²
x1 = ( - 3 + 5) / 2 = 1
x2 = ( - 3 - 5) / 2 = - 4
5) x² - 5x + 4 = 0
D = b² - 4ac = 25 - 4×4 = 9 = 3²
x1 =( 5 + 3) / 2 = 4
x2 = ( 5 - 3) / 2 = 1
6) x² - 4x + 3 = 0
D = b - 4ac = 16 - 4×3 = 4 = 2²
x1 = ( 4 + 2) / 2 = 3
x2 = ( 4 - 2) / 2 = 1
7) 2x² + x - 3x - 4 = 0
2x² - 2x - 4 = 0
x² - x - 2 = 0
D = b² - 4ac = 1 - 4×(-2) = 9 = 3²
x1 = ( 1 + 3) / 2 = 2
x2 = ( 1 - 3) / 2 = - 1
8) 2x² - 3x - 4x + 3 = 0
2x² - 7x + 3 = 0
D = b²- 4ac = 49 - 8×3 = 25 = 5²
x1 = ( 7 + 5) / 4 = 3
x2 = ( 7 - 5)/ 4 = 0,5