ОГЭ 2026. Задание №9. Уравнения нового и старого банка ФИПИ.

Решение:

Блок №1. Линейные уравнения:

1. \( 3x + 3 = 5x \)
   \( 2x = 3 \)
   \( x = \frac{3}{2} \)


2. \( 6x + 1 = -4x \)
   \( 10x = -1 \)
   \( x = -\frac{1}{10} \)


3. \( -4x - 9 = 6x \)
   \( -9 = 10x \)
   \( x = -\frac{9}{10} \)


4. \( x + 3 = -9x \)
   \( 10x = -3 \)
   \( x = -\frac{3}{10} \)


5. \( 4(x - 8) = -5 \)
   \( 4x - 32 = -5 \)
   \( 4x = 27 \)
   \( x = \frac{27}{4} \)


6. \( 5(x + 4) = -9 \)
   \( 5x + 20 = -9 \)
   \( 5x = -29 \)
   \( x = -\frac{29}{5} \)


7. \( 5(x + 9) = -8 \)
   \( 5x + 45 = -8 \)
   \( 5x = -53 \)
   \( x = -\frac{53}{5} \)


8. \( 10(x + 2) = -7 \)
   \( 10x + 20 = -7 \)
   \( 10x = -27 \)
   \( x = -\frac{27}{10} \)


9. \( 8 + 7x = 9x + 4 \)
   \( 4 = 2x \)
   \( x = 2 \)


10. \( 9 + 8x = 6x - 2 \)
    \( 2x = -11 \)
    \( x = -\frac{11}{2} \)


11. \( -5 + 9x = 10x + 4 \)
    \( -9 = x \)
    \( x = -9 \)


12. \( -4 + 7x = 8x + 1 \)
    \( -5 = x \)
    \( x = -5 \)

Блок №2. Квадратные уравнения:

1. \( x^2 = 5x \)
   \( x^2 - 5x = 0 \)
   \( x(x - 5) = 0 \)
   \( x_1 = 0, x_2 = 5 \)


2. \( 2x^2 = 8x \)
   \( 2x^2 - 8x = 0 \)
   \( 2x(x - 4) = 0 \)
   \( x_1 = 0, x_2 = 4 \)


3. \( 3x^2 = 9x \)
   \( 3x^2 - 9x = 0 \)
   \( 3x(x - 3) = 0 \)
   \( x_1 = 0, x_2 = 3 \)


4. \( 4x^2 = 20x \)
   \( 4x^2 - 20x = 0 \)
   \( 4x(x - 5) = 0 \)
   \( x_1 = 0, x_2 = 5 \)


5. \( x^2 - 9 = 0 \)
   \( x^2 = 9 \)
   \( x = \pm 3 \)


6. \( x^2 - 121 = 0 \)
   \( x^2 = 121 \)
   \( x = \pm 11 \)


7. \( x^2 - 16 = 0 \)
   \( x^2 = 16 \)
   \( x = \pm 4 \)


8. \( x^2 - 49 = 0 \)
   \( x^2 = 49 \)
   \( x = \pm 7 \)


9. \( 5x^2 - 10x = 0 \)
   \( 5x(x - 2) = 0 \)
   \( x_1 = 0, x_2 = 2 \)


10. \( 3x^2 - 9x = 0 \)
    \( 3x(x - 3) = 0 \)
    \( x_1 = 0, x_2 = 3 \)


11. \( 4x^2 - 16x = 0 \)
    \( 4x(x - 4) = 0 \)
    \( x_1 = 0, x_2 = 4 \)


12. \( 5x^2 + 15x = 0 \)
    \( 5x(x + 3) = 0 \)
    \( x_1 = 0, x_2 = -3 \)


13. \( x^2 - 6x + 5 = 0 \)
    \( D = (-6)^2 - 4 \cdot 1 \cdot 5 = 36 - 20 = 16 \)
    \( x_{1,2} = \frac{6 \pm \sqrt{16}}{2} = \frac{6 \pm 4}{2} \)
    \( x_1 = \frac{10}{2} = 5, x_2 = \frac{2}{2} = 1 \)


14. \( x^2 - 7x + 10 = 0 \)
    \( D = (-7)^2 - 4 \cdot 1 \cdot 10 = 49 - 40 = 9 \)
    \( x_{1,2} = \frac{7 \pm \sqrt{9}}{2} = \frac{7 \pm 3}{2} \)
    \( x_1 = \frac{10}{2} = 5, x_2 = \frac{4}{2} = 2 \)