Question

сокращение рациональных дробей
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Answer 1

1. $\frac{4 {a}^{4} b}{7ab} = \frac{4 {a}^{3} }{7}$

2. $\frac{xyk}{ {k}^{3} x {y}^{3} } = \frac{1}{ {y}^{2} {k}^{2} }$

3. $\frac{ - 6 {(m + n)}^{2} }{3(m + n)} = \frac{ - 2(m + n)(m + n)}{m + n} = - 2(m + n) = - 2m - 2n$

4. $\frac{6a + 9b}{3a} = \frac{3(2a + 3b)}{3a} = \frac{2a + 3b}{a}$

5. $\frac{2a - 7b}{4a - 14b} = \frac{2a - 7b}{2(2a - 7b)} = \frac{1}{2}$

6. $\frac{3x + 3y}{9xy} = \frac{3(x + y)}{9xy} = \frac{x + y}{3xy}$

7. $\frac{ {a}^{2} - 12a + 36 }{ {a}^{2} - 36} = \frac{ {(a - 6)}^{2} }{ {a}^{2} - {6}^{2} } = \frac{(a - 6)(a - 6)}{(a - 6)(a + 6)} = \frac{a - 6}{a + 6}$

8.

$\frac{9 {x}^{2} + 6x + 1 }{6 {x}^{2} - x - 1} = \frac{ {(3x + 1)}^{2} }{6(x - \frac{1}{2} )(x + \frac{1}{3} )} = \frac{(3x + 1)(3x + 1)}{2(x - \frac{1}{2}) \times 3(x + \frac{1}{3} )} = \frac{(3x + 1)(3x + 1)}{(2x - 1)(3x + 1)} = \frac{3x + 1}{2x - 1}$

Разложение на множители:

$a {x}^{2} + bx + c = a (x - x_1 )(x - x_2)$

$9 {x}^{2} + 6x + 1= 6(x - \frac{1}{2} )(x + \frac{1}{3})$

$D = {b}^{2} - 4ac = {( - 1)}^{2} - 4 \times 6 \times ( - 1) = 1 + 24 = 25 = {5}^{2}$

$x_1 = \frac{ - b + \sqrt{D} }{2a} = \frac{1 + 5}{2 \times 6} = \frac{6}{12} = \frac{1}{2}$

$x_2 = \frac{ - b - \sqrt{D} }{2a} = \frac{1 - 5}{2 \times 6} = \frac{ - 4}{12} = - \frac{ 1}{3}$