Question

Сократите дробь : 1) a^2+5a+6/a^2+4a+4 2) x^2+3x+2/x^2+6x+5 3) m^2+2m+1/m^2+8m+7

Answer 1

1) $\frac{ a^{2} + 5a + 6}{a^{2} + 4a + 4}$

a² + 5a + 6 = 0
D = 5² - 4 * 6 * 1 = 25 - 24 = 1
a₁ = (-5 + √1) / 2*1 = (-5 + 1) / 2 = -4 / 2 = -2
a₂ = (-5 - √1) / 2*1 = (-5 - 1) / 2 = -6 / 2 = -3
a² + 5a + 6 = (a + 2)(a + 3)

a² + 4a + 4 = 0
D = 4² - 4*4*1 = 16 - 16 = 0
a = -4 / 2*1 = -2
a² + 4a + 4 = (a + 2)(a + 2)

$\frac{ a^{2} + 5a + 6}{a^{2} + 4a + 4} = \frac{ (a+2)(a+3)}{(a+2)(a+2)} = \frac{a+3}{a+2}$
ответ: a+3 / a+2

2) $\frac{ x^{2} + 3x + 2}{x^{2} + 6x + 5}$

x² + 3x + 2 = 0
D = 9 - 8 = 1
x₁ = (-3+1) / 2 = -2 / 2 = -1
x₂ = -4 / 2 = -2
x² + 3x + 2 = (x + 1)(x + 2)

x² + 6x + 5 = 0
D = 36 - 20 = 16
x₁ = (-6 + 4) / 2 = -2 / 2 = -1
x₂ = -10 / 2 = -5
x² + 6x + 5 = (x + 1)(x + 5)

$\frac{ x^{2} + 3x + 2}{x^{2} + 6x + 5} = \frac{(x + 1)(x + 2)}{(x + 1)(x + 5)} = \frac{x+2}{x+5}$
ответ: x+2 / x+5

3) $\frac{ m^{2} + 2m + 1}{m^{2} + 8m + 7}$

m² + 2m + 1 = 0
D = 4 - 4 = 0
m = -2/2 = -1
m² + 2m + 1 = (m + 1)(m + 1)

m² + 8m + 7 = 0
D = 64 - 28 = 36
m₁ = (-8+6) / 2 = -2 / 2 = -1
m₂ = (-8-6) / 2 = -14 / 2 = -7

$\frac{ m^{2} + 2m + 1 }{ m^{2} + 8m + 7 } = \frac{(m+1)(m+1)}{(m+1)(m+7)} = \frac{m+1}{m+7}$
ответ: m+1 / m+7