Решени задаци.
Текст задатака објашњених у видео лекцији:
Пр.1) Решити једначине:
а) $\left( {4x - 3} \right)\left( {3x + 4} \right) - \left( {2x + 1} \right)\left( {6x - 1} \right) = 1$
б) ${\left( {3 - 5x} \right)^2} + {\left( {1 + 12x} \right)^2} = {\left( {13x - 2} \right)^2} + 6$
в) $\left( {2x - 3} \right)\left( {2x + 3} \right) - {\left( {2x - 1} \right)^2} = 8 + 6x$
Пр.2) Решити следеће једначине:
а) $\left( {x + 3} \right):\left( {1 - x} \right) = - 2$
б) $\left( {x + 1} \right):\left( {x + 3} \right) = \left( {x - 3} \right):\left( {x - 2} \right)$
Пр.1)
а) $\left( {4x - 3} \right)\left( {3x + 4} \right) - \left( {2x + 1} \right)\left( {6x - 1} \right) = 1$
$12{x^2} + 16x - 9x - 12 - \left( {12{x^2} - 2x + 6x - 1} \right) = 1 $
$ 12{x^2} + 7x - 12 - 12{x^2} + 2x - 6x + 1 = 1 $
$ 3x - 11 = 1 $
$ 3x = 12 $
$ x = 4 $
б) ${\left( {3 - 5x} \right)^2} + {\left( {1 + 12x} \right)^2} = {\left( {13x - 2} \right)^2} + 6$
${3^2} - 2 \cdot 3 \cdot 5x + {\left( {5x} \right)^2} + 1 + 24x + 144{x^2} = 169{x^2} - 52x + 4 + 6 $
$9 - 30x + 25{x^2} + 1 + 24x + 144{x^2} = 169{x^2} - 52x + 10 $
$10 - 6x + 169{x^2} = 169{x^2} - 52x + 10 $
$ - 6x + 169{x^2} - 169{x^2} + 52x = 10 - 10 $
$46x = 0 $
$x = \frac{0}{{46}} $
$x = 0 $
в) $\left( {2x - 3} \right)\left( {2x + 3} \right) - {\left( {2x - 1} \right)^2} = 8 + 6x$
${\left( {2x} \right)^2} - {3^2} - \left( {4{x^2} - 4x + 1} \right) = 8 + 6x$
$4{x^2} - 9 - 4{x^2} + 4x - 1 = 8 + 6x$
$4x - 10 = 8 + 6x$
$4x - 6x = 10 + 8$
$ - 2x = 18$
$x = - 9$
Пр.2)
а) $\left( {x + 3} \right):\left( {1 - x} \right) = - 2$
$\left( {x + 3} \right):\left( {1 - x} \right) = - 2:1$
$1 \cdot \left( {x + 3} \right) = - 2 \cdot \left( {1 - x} \right)$
$x + 3 = - 2 + 2x$
$x - 2x = - 2 - 3$
$ - x = - 5$
$x = 5$
б) $\left( {x + 1} \right):\left( {x + 3} \right) = \left( {x - 3} \right):\left( {x - 2} \right)$
$\left( {x + 1} \right)\left( {x - 2} \right) = \left( {x + 3} \right)\left( {x - 3} \right)$
${x^2} - 2x + x - 2 = {x^2} - {3^2}$
${x^2} - x - 2 = {x^2} - 9$
${x^2} - x - {x^2} = 2 - 9$
$ - x = - 7$
$x = 7$