ГДЗ по алгебре 8 класс Макарычев Задание 542

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Год:2021
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Задание 542

\[\boxed{\text{542\ (542).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[\textbf{а)}\ 5x^{2} = 9x + 2\]

\[5x^{2} - 9x - 2 = 0\]

\[D = 81 + 40 = 121\]

\[x_{1,2} = \frac{9 \pm \sqrt{121}}{10} = \frac{9 \pm 11}{10}\]

\[x_{1} = 2;\ \ x_{2} = - 0,2.\]

\[\textbf{б)} - x^{2} = 5x - 14\]

\[- x^{2} - 5x + 14 = 0\]

\[x^{2} + 5x - 14 = 0\]

\[D = 25 + 56 = 81\]

\[x_{1,2} = \frac{- 5 \pm \sqrt{81}}{2 \cdot 1} = \frac{- 5 \pm 9}{2}\]

\[x_{1} = - 7;\ \ x_{2} = 2.\]

\[\textbf{в)}\ 6x + 9 = x^{2}\]

\[x^{2} - 6x - 9 = 0\]

\[D_{1} = 3^{2} + 9 = 9 + 9 = 18\]

\[x_{1,2} = 3 \pm \sqrt{18} = 3 \pm 3\sqrt{2}\]

\[x_{1} = 3 + 3\sqrt{2};\ \ x_{2} = 3 - 3\sqrt{2}.\]

\[\textbf{г)}\ z - 5 = z^{2} - 25\]

\[z^{2} - z - 20 = 0\]

\[D = 1 + 80 = 81\]

\[z_{1,2} = \frac{1 \pm \sqrt{81}}{2} = \frac{1 \pm 9}{2}\]

\[z_{1} = 5;\ \ z_{2} = - 4.\]

\[\textbf{д)}\ y^{2} = 52y - 576\]

\[y^{2} - 52y + 576 = 0\]

\[D_{1} = 26^{2} - 576 =\]

\[= 676 - 576 = 100\]

\[y = 26 \pm 10\]

\[y_{1} = 26 - 10 = 16;\]

\[y_{2} = 26 + 10 = 36.\]

\[\textbf{е)}\ 15y^{2} - 30 = 22y + 7\]

\[15y^{2} - 22y - 37 = 0\]

\[D_{1} = 11^{2} + 37 \cdot 15 =\]

\[= 121 + 555 = 676\]

\[y_{1,2} = \frac{11 \pm \sqrt{676}}{15} = \frac{11 \pm 26}{15}\]

\[y_{1} = \frac{37}{15} = 2\frac{7}{15};\ \ y_{2} = - 1.\]

\[\textbf{ж)}\ 25p^{2} = 10p - 1\]

\[25p^{2} - 10p + 1 = 0\]

\[D_{1} = 5^{2} - 1 \cdot 25 = 25 - 25 = 0\]

\[p = \frac{5}{25} = \frac{1}{5} = 0,2.\]

\[\textbf{з)}\ 299x^{2} + 100x = 500 - 101x^{2}\]

\[400x^{2} + 100x - 500 = 0\ \ \ |\ :100\]

\[4x^{2} + x - 5 = 0\]

\[D = 1 + 80 = 81\]

\[x_{1,2} = \frac{- 1 \pm \sqrt{81}}{8} = \frac{- 1 \pm 9}{8}\]

\[x_{1} = 1;\ \ x_{2} = - \frac{5}{4} = - 1,25.\ \]

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