Ответы к странице 185
752. Можно ли разложить на линейные множители квадратный трехчлен:
1) $x^2 — 12x + 6$;
2) $3x^2 — 8x + 6$;
3) $2a^2 — 8a + 8$;
4) $-6b^2 + b + 12$?
Решение:
1) $x^2 — 12x + 6 = 0$
$D = b^2 — 4ac = (-12)^2 — 4 * 1 * 6 = 144 — 24 = 120 > 0$ − значит разложить на линейные множители можно2) $3x^2 — 8x + 6 = 0$
$D = b^2 — 4ac = (-8)^2 — 4 * 3 * 6 = 64 — 72 = -8 < 0$ − значит разложить на линейные множители можно3) $2a^2 — 8a + 8 = 0$
$D = b^2 — 4ac = (-8)^2 — 4 * 2 * 8 = 64 — 64 = 0$ − значит разложить на линейные множители можно4) $-6b^2 + b + 12 = 0$
$D = b^2 — 4ac = 1^2 — 4 * (-6) * 12 = 1 + 288 = 289 > $ − значит разложить на линейные множители можно
753. Разложите на линейные множители квадратный трехчлен:
1) $x^2 — 7x + 12$;
2) $x^2 + 8x + 15$;
3) $x^2 — 3x — 10$;
4) $-x^2 — 5x — 6$;
5) $-x^2 + x + 2$;
6) $6x^2 — 5x — 1$;
7) $4x^2 + 3x — 22$;
8) $-3a^2 + 8a + 3$;
9) $frac{1}{6}b^2 — frac{5}{6}b + 1$;
10) $-2x^2 — 0,5x + 1,5$;
11) $0,4x^2 — 2x + 2,5$;
12) $-1,2m^2 + 2,6m — 1$.
Решение:
1) $x^2 — 7x + 12 = 0$
$D = b^2 — 4ac = (-7)^2 — 4 * 1 * 12 = 49 — 48 = 1 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{7 + sqrt{1}}{2 * 1} = frac{7 + 1}{2} = frac{8}{2} = 4$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{7 — sqrt{1}}{2 * 1} = frac{7 — 1}{2} = frac{6}{2} = 3$
Ответ:$x^2 — 7x + 12 = (x — 3)(x — 4)$2) $x^2 + 8x + 15 = 0$
$D = b^2 — 4ac = 8^2 — 4 * 1 * 15 = 64 — 60 = 4 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-8 + sqrt{4}}{2 * 1} = frac{-8 + 2}{2} = frac{-6}{2} = -3$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-8 — sqrt{4}}{2 * 1} = frac{-8 -2}{2} = frac{-10}{2} = -5$
$x^2 + 8x + 15 = (x — (-5))(x — (-3)) = (x + 5)(x + 3)$
Ответ: $x^2 + 8x + 15 = (x + 5)(x + 3)$3) $x^2 — 3x — 10 = 0$
$D = b^2 — 4ac = (-3)^2 — 4 * 1 * (-10) = 9 + 40 = 49 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{3 + sqrt{49}}{2 * 1} = frac{3 + 7}{2} = frac{10}{2} = 5$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{3 — sqrt{49}}{2 * 1} = frac{3 — 7}{2} = frac{-4}{2} = -2$
$x^2 — 3x — 10 = (x — 5)(x — (-2)) = (x — 5)(x + 2)$
Ответ: $x^2 — 3x — 10 = (x — 5)(x + 2)$4) $-x^2 — 5x — 6 = 0$
$D = b^2 — 4ac = (-5)^2 — 4 * (-1) * (-6) = 25 — 24 = 1 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{5 + sqrt{1}}{2 * (-1)} = frac{5 + 1}{-2} = frac{6}{-2} = -3$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{5 — sqrt{1}}{2 * (-1)} = frac{5 — 1}{-2} = frac{4}{-2} = -2$
$-x^2 — 5x — 6 = -(x — (-3))(x — (-2)) = -(x + 3)(x + 2)$
Ответ: $-x^2 — 5x — 6 = -(x + 3)(x + 2)$5) $-x^2 + x + 2 = 0$
$D = b^2 — 4ac = 1^2 — 4 * (-1) * 2 = 1 + 8 = 9 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-1 + sqrt{9}}{2 * (-1)} = frac{-1 + 3}{-2} = frac{2}{-2} = -1$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-1 — sqrt{9}}{2 * (-1)} = frac{-1 — 3}{-2} = frac{-4}{-2} = 2$
$-x^2 + x + 2 = -(x — (-1))(x — 2) = -(x + 1)(x — 2) = (x + 1)(2 — x)$
Ответ: $-x^2 + x + 2 = (x + 1)(2 — x)$6) $6x^2 — 5x — 1 = 0$
$D = b^2 — 4ac = (-5)^2 — 4 * 6 * (-1) = 25 + 24 = 49 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{5 + sqrt{49}}{2 * 6} = frac{5 + 7}{12} = frac{12}{12} = 1$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{5 — sqrt{49}}{2 * 6} = frac{5 — 7}{12} = frac{-2}{12} = -frac{1}{6}$
$6x^2 — 5x — 1 = 6(x — (-frac{1}{6}))(x — 1) = 6(x + frac{1}{6})(x — 1) = (6x + 1)(x — 1)$
Ответ: $6x^2 — 5x — 1 = (6x + 1)(x — 1)$7) $4x^2 + 3x — 22 = 0$
$D = b^2 — 4ac = 3^2 — 4 * 4 * (-22) = 9 + 352 = 361 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-3 + sqrt{361}}{2 * 4} = frac{-3 + 19}{8} = frac{16}{8} = 2$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-3 — sqrt{361}}{2 * 4} = frac{-3 — 19}{8} = frac{-22}{8} = -frac{11}{4}$
$4x^2 + 3x — 22 = 4(x — (-frac{11}{4}))(x — 2) = 4(x + frac{11}{4})(x — 2) = (4x + 11)(x — 2)$
Ответ: $4x^2 + 3x — 22 = (4x + 11)(x — 2)$8) $-3a^2 + 8a + 3 = 0$
$D = b^2 — 4ac = 8^2 — 4 * (-3) * 3 = 64 + 36 = 100 > 0$
$a_1 = frac{-b + sqrt{D}}{2a} = frac{-8 + sqrt{100}}{2 * (-3)} = frac{-8 + 10}{-6} = frac{2}{-6} = -frac{1}{3}$
$a_2 = frac{-b — sqrt{D}}{2a} = frac{-8 — sqrt{100}}{2 * (-3)} = frac{-8 — 10}{-6} = frac{-18}{-6} = 3$
$-3a^2 + 8a + 3 = -3(a — (-frac{1}{3}))(a — 3) = -3(a + frac{1}{3})(a — 3) = (3a + 1)(3 — a)$
Ответ: $-3a^2 + 8a + 3 = (3a + 1)(3 — a)$9) $frac{1}{6}b^2 — frac{5}{6}b + 1 = 0$
$D = b^2 — 4ac = (-frac{5}{6})^2 — 4 * frac{1}{6} * 1 = frac{25}{36} — frac{2}{3} = frac{25 — 24}{36} = frac{1}{36} > 0$
$b_1 = frac{-b + sqrt{D}}{2a} = frac{frac{5}{6} + sqrt{frac{1}{36}}}{2 * frac{1}{6}} = frac{frac{5}{6} + frac{1}{6}}{frac{1}{3}} = frac{1}{frac{1}{3}} = 3$
$b_2 = frac{-b — sqrt{D}}{2a} = frac{frac{5}{6} — sqrt{frac{1}{36}}}{2 * frac{1}{6}} = frac{frac{5}{6} — frac{1}{6}}{frac{1}{3}} = frac{frac{4}{6}}{frac{1}{3}} = frac{frac{2}{3}}{frac{1}{3}} = 2$
$frac{1}{6}b^2 — frac{5}{6}b + 1 = frac{1}{6}(b — 3)(b — 2) = frac{1}{3}(b — 3) * frac{1}{2}(b — 2) = (frac{1}{3}b — 1)(frac{1}{2}b — 1)$
Ответ: $frac{1}{6}b^2 — frac{5}{6}b + 1 = (frac{1}{3}b — 1)(frac{1}{2}b — 1)$10) $-2x^2 — 0,5x + 1,5 = 0$
$D = b^2 — 4ac = (-0,5)^2 — 4 * (-2) * 1,5 = 0,25 + 12 = 12,25 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{0,5 + sqrt{12,25}}{2 * (-2)} = frac{0,5 + 3,5}{-4} = frac{4}{-4} = -1$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{0,5 — sqrt{12,25}}{2 * (-2)} = frac{0,5 — 3,5}{-4} = frac{3}{4}$
$-2x^2 — 0,5x + 1,5 = -2(x — (-1))(x — frac{3}{4}) = -2(x + 1)(x — frac{3}{4}) = (x + 1)(frac{3}{2} — x) = (x + 1)(1,5 — x)$
Ответ: $-2x^2 — 0,5x + 1,5 = (x + 1)(1,5 — x)$11) $0,4x^2 — 2x + 2,5 = 0$
$D = b^2 — 4ac = (-2)^2 — 4 * 0,4 * 2,5 = 4 — 4 = 0$
$x = frac{-b + sqrt{D}}{2a} = frac{2 + sqrt{0}}{2 * 0,4} = frac{2}{0,8} = frac{20}{8} = frac{5}{2} = 2,5$
$0,4x^2 — 2x + 2,5 = 0,4(x — 2,5)(x — 2,5) = 0,4(x — 2,5)^2$
Ответ: $0,4x^2 — 2x + 2,5 = 0,4(x — 2,5)^2$12) $-1,2m^2 + 2,6m — 1 = 0$
$D = b^2 — 4ac = 2,6^2 — 4 * (-1,2) * (-1) = 6,76 — 4,8 = 1,96 > 0$
$m_1 = frac{-b + sqrt{D}}{2a} = frac{-2,6 + sqrt{1,96}}{2 * (-1,2)} = frac{-2,6 + 1,4}{-2,4} = frac{-1,2}{-2,4} = 0,5$
$m_2 = frac{-b — sqrt{D}}{2a} = frac{-2,6 — sqrt{1,96}}{2 * (-1,2)} = frac{-2,6 — 1,4}{-2,4} = frac{-4}{2,4} = frac{40}{24} = frac{5}{3}$
$-1,2m^2 + 2,6m — 1 = 0 = -1,2(m — 0,5)(m — frac{5}{3}) = (m — 0,5)(frac{12}{10} * frac{5}{3} -1,2m) = (m — 0,5)(frac{6}{5} * frac{5}{3} -1,2m) = (m — 0,5)(2 -1,2m)$
Ответ: $-1,2m^2 + 2,6m — 1 = (m — 0,5)(2 -1,2m)$
754. Разложите на линейные множители квадратный трехчлен:
1) $x^2 — 3x — 18$;
2) $x^2 + 5x — 14$;
3) $-x^2 + 3x + 4$;
4) $5x^2 + 8x — 4$;
5) $2a^2 — 3a + 1$;
6) $4b^2 — 11b — 3$;
7) $-frac{1}{4}x^2 — 2x — 3$;
8) $0,3m^2 — 3m + 7,5$;
9) $x^2 — 2x — 2$.
Решение:
1) $x^2 — 3x — 18 = 0$
$D = b^2 — 4ac = (-3)^2 — 4 * 1 * (-18) = 9 + 72 = 81 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{3 + sqrt{81}}{2 * 1} = frac{3 + 9}{2} = frac{12}{2} = 6$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{3 — sqrt{81}}{2 * 1} = frac{3 — 9}{2} = frac{-6}{2} = -3$
$x^2 — 3x — 18 = (x — 6)(x — (-3)) = (x — 6)(x + 3)$
Ответ:$x^2 — 3x — 18 = (x — 6)(x + 3)$2) $x^2 + 5x — 14 = 0$
$D = b^2 — 4ac = 5^2 — 4 * 1 * (-14) = 25 + 56 = 81 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-5 + sqrt{81}}{2 * 1} = frac{-5 + 9}{2} = frac{4}{2} = 2$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-5 — sqrt{81}}{2 * 1} = frac{-5 — 9}{2} = frac{-14}{2} = -7$
$x^2 + 5x — 14 = (x — 2)(x — (-7)) = (x — 2)(x + 7)$
Ответ: $x^2 + 5x — 14 = (x — 2)(x + 7)$3) $-x^2 + 3x + 4 = 0$
$D = b^2 — 4ac = 3^2 — 4 * (-1) * 4 = 9 + 16 = 25 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-3 + sqrt{25}}{2 * (-1)} = frac{-3 + 5}{-2} = frac{2}{-2} = -1$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-3 — sqrt{25}}{2 * (-1)} = frac{-3 — 5}{-2} = frac{-8}{-2} = 4$
$-x^2 + 3x + 4 = -(x — (-1))(x — 4) = (x + 1)(4 — x)$
Ответ: $-x^2 + 3x + 4 = (x + 1)(4 — x)$4) $5x^2 + 8x — 4 = 0$
$D = b^2 — 4ac = 8^2 — 4 * 5 * (-4) = 64 + 80 = 144 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-8 + sqrt{144}}{2 * 5} = frac{-8 + 12}{10} = frac{4}{10} = 0,4$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-8 — sqrt{144}}{2 * 5} = frac{-8 — 12}{10} = frac{-20}{10} = -2$
$5x^2 + 8x — 4 = 5(x — 0,4)(x — (-2)) = (5x — 2)(x + 2)$
Ответ: $5x^2 + 8x — 4 = (5x — 2)(x + 2)$5) $2a^2 — 3a + 1 = 0$
$D = b^2 — 4ac = (-3)^2 — 4 * 2 * 1 = 9 + 8 = 1 > 0$
$a_1 = frac{-b + sqrt{D}}{2a} = frac{3 + sqrt{1}}{2 * 2} = frac{3 + 1}{4} = frac{4}{4} = 1$
$a_2 = frac{-b — sqrt{D}}{2a} = frac{3 — sqrt{1}}{2 * 2} = frac{3 — 1}{4} = frac{2}{4} = 0,5$
$2a^2 — 3a + 1 = 2(a — 1)(a — 0,5) = (a — 1)(2a — 1)$
Ответ: $2a^2 — 3a + 1 = (a — 1)(2a — 1)$6) $4b^2 — 11b — 3 = 0$
$D = b^2 — 4ac = (-11)^2 — 4 * 4 * (-3) = 121 + 48 = 169 > 0$
$b_1 = frac{-b + sqrt{D}}{2a} = frac{11 + sqrt{169}}{2 * 4} = frac{11 + 13}{8} = frac{24}{8} = 3$
$b_2 = frac{-b — sqrt{D}}{2a} = frac{11 — sqrt{169}}{2 * 4} = frac{11 — 13}{8} = frac{-2}{8} = -frac{1}{4}$
$4b^2 — 11b — 3 = 4(b — 3)(b — (-frac{1}{4})) = 4(b — 3)(b + frac{1}{4}) = (b — 3)(4b + 1)$
Ответ: $4b^2 — 11b — 3 = (b — 3)(4b + 1)$7) $-frac{1}{4}x^2 — 2x — 3 = 0$
$D = b^2 — 4ac = (-2)^2 — 4 * (-frac{1}{4}) * (-3) = 4 — 3 = 1 > 0$
$a_1 = frac{-b + sqrt{D}}{2a} = frac{2 + sqrt{1}}{2 * (-frac{1}{4})} = frac{2 + 1}{-frac{1}{2}} = frac{3}{-frac{1}{2}} = -6$
$a_2 = frac{-b — sqrt{D}}{2a} = frac{2 — sqrt{1}}{2 * (-frac{1}{4})} = frac{2 — 1}{-frac{1}{2}} = frac{1}{-frac{1}{2}} = -2$
$-frac{1}{4}x^2 — 2x — 3 = -frac{1}{4}(x — (-6))(x — (-2)) = -frac{1}{4}(x + 6)(x + 2)$
Ответ: $-frac{1}{4}x^2 — 2x — 3 = -frac{1}{4}(x + 6)(x + 2)$8) $0,3m^2 — 3m + 7,5 = 0$
$D = b^2 — 4ac = (-3)^2 — 4 * 0,3 * 7,5 = 9 — 9 = 0$
$m = frac{-b + sqrt{D}}{2a} = frac{3 + sqrt{0}}{2 * 0,3} = frac{3}{0,6} = frac{30}{6} = 5$
$0,3m^2 — 3m + 7,5 = 0,3(m — 5)(m — 5) = 0,3(m — 5)^2$
Ответ: $0,3m^2 — 3m + 7,5 = 0,3(m — 5)^2$9) $x^2 — 2x — 2 = 0$
$D = b^2 — 4ac = (-2)^2 — 4 * 1 * (-2) = 4 + 8 = 12 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{2 + sqrt{12}}{2 * 1} = frac{2 + sqrt{4 * 3}}{2} = frac{2 + 2sqrt{3}}{2} = frac{2(1 + sqrt{3})}{2} = 1 + sqrt{3}$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{2 — sqrt{12}}{2 * 1} = frac{2 — sqrt{4 * 3}}{2} = frac{2 — 2sqrt{3}}{2} = frac{2(1 — sqrt{3})}{2} = 1 — sqrt{3}$
$x^2 — 2x — 2 = (x — (1 + sqrt{3}))(x — (1 — sqrt{3})) = (x — 1 — sqrt{3})(x — 1 + sqrt{3})$
Ответ:$x^2 — 2x — 2 = (x — 1 — sqrt{3})(x — 1 + sqrt{3})$
755. Сократите дробь:
1) $frac{x^2 + x — 6}{x + 3}$;
2) $frac{x — 4}{x^2 — 10x + 24}$;
3) $frac{3x — 15}{x^2 — x — 20}$;
4) $frac{x^2 — 3x + 2}{6x — 6}$;
5) $frac{x^2 — 7x +12}{x^2 — 3x}$;
6) $frac{x^2 + 4x}{x^2 + 2x — 8}$.
Решение:
1) $frac{x^2 + x — 6}{x + 3}$
$x^2 + x — 6 = 0$
$D = b^2 — 4ac = 1^2 — 4 * 1 * (-6) = 1 + 24 = 25 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-1 + sqrt{25}}{2 * 1} = frac{-1 + 5}{2} = frac{4}{2} = 2$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-1 — sqrt{25}}{2 * 1} = frac{-1 — 5}{2} = frac{-6}{2} = -3$
$x^2 + x — 6 = (x — 2)(x — (-3)) = (x — 2)(x + 3)$
$frac{x^2 + x — 6}{x + 3} = frac{(x — 2)(x + 3)}{x + 3} = x — 2$
Ответ: x − 22) $frac{x — 4}{x^2 — 10x + 24}$
$x^2 — 10x + 24 = 0$
$D = b^2 — 4ac = (-10)^2 — 4 * 1 * 24 = 100 + 96 = 4 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{10 + sqrt{4}}{2 * 1} = frac{10 + 2}{2} = frac{12}{2} = 6$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{10 — sqrt{4}}{2 * 1} = frac{10 — 2}{2} = frac{8}{2} = 4$
$x^2 — 10x + 24 = (x — 6)(x — 4)$
$frac{x — 4}{x^2 — 10x + 24} = frac{x — 4}{(x — 6)(x — 4)} = frac{1}{x — 6}$
Ответ: $frac{1}{x — 6}$3) $frac{3x — 15}{x^2 — x — 20}$
$x^2 — x — 20 = 0$
$D = b^2 — 4ac = (-1)^2 — 4 * 1 * (-20) = 1 + 80 = 81 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{1 + sqrt{81}}{2 * 1} = frac{1 + 9}{2} = frac{10}{2} = 5$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{1 — sqrt{81}}{2 * 1} = frac{1 — 9}{2} = frac{-8}{2} = -4$
$x^2 — x — 20 = (x — 5)(x — (-4)) = (x — 5)(x + 4)$
$frac{3x — 15}{x^2 — x — 20} = frac{3(x — 5)}{(x — 5)(x + 4)} = frac{3}{x + 4}$
Ответ: $frac{3}{x + 4}$4) $frac{x^2 — 3x + 2}{6x — 6}$
$x^2 — 3x + 2 = 0$
$D = b^2 — 4ac = (-3)^2 — 4 * 1 * 2 = 9 — 8 = 1 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{3 + sqrt{1}}{2 * 1} = frac{3 + 1}{2} = frac{4}{2} = 2$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{3 — sqrt{1}}{2 * 1} = frac{3 — 1}{2} = frac{2}{2} = 1$
$x^2 — 3x + 2 = (x — 2)(x — 1)$
$frac{x^2 — 3x + 2}{6x — 6} = frac{(x — 2)(x — 1)}{6(x — 1)} = frac{x — 2}{6}$
Ответ: $frac{x — 2}{6}$5) $frac{x^2 — 7x + 12}{x^2 — 3x}$
$x^2 — 7x + 12 = 0$
$D = b^2 — 4ac = (-7)^2 — 4 * 1 * 12 = 49 — 48 = 1 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{7 + sqrt{1}}{2 * 1} = frac{7 + 1}{2} = frac{8}{2} = 4$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{7 — sqrt{1}}{2 * 1} = frac{7 — 1}{2} = frac{6}{2} = 3$
$x^2 — 7x + 12 = (x — 4)(x — 3)$
$frac{x^2 — 7x + 12}{x^2 — 3x} = frac{(x — 4)(x — 3)}{x(x — 3)} = frac{x — 4}{x}$
Ответ: $frac{x — 4}{x}$6) $frac{x^2 + 4x}{x^2 + 2x — 8}$
$x^2 + 2x — 8 = 0$
$D = b^2 — 4ac = 2^2 — 4 * 1 * (-8) = 4 + 32 = 36 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-2 + sqrt{36}}{2 * 1} = frac{-2 + 6}{2} = frac{4}{2} = 2$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-2 — sqrt{36}}{2 * 1} = frac{-2 — 6}{2} = frac{-8}{2} = -4$
$x^2 + 2x — 8 = (x — 2)(x — (-4)) = (x — 2)(x + 4)$
$frac{x^2 + 4x}{x^2 + 2x — 8} = frac{x(x + 4)}{(x — 2)(x + 4)} = frac{x}{x — 2}$
Ответ: $frac{x}{x — 2}$
756. Сократите дробь:
1) $frac{x^2 — 6x + 5}{x — 5}$;
2) $frac{2x + 12}{x^2 + 3x — 18}$;
3) $frac{x^2 + 9x + 14}{x^2 + 7x}$.
Решение:
1) $frac{x^2 — 6x + 5}{x — 5}$
$x^2 — 6x + 5 = 0$
$D = b^2 — 4ac = (-6)^2 — 4 * 1 * 5 = 36 — 20 = 16 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{6 + sqrt{16}}{2 * 1} = frac{6 + 4}{2} = frac{10}{2} = 5$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{6 — sqrt{16}}{2 * 1} = frac{6 — 4}{2} = frac{2}{2} = 1$
$x^2 — 6x + 5 = (x — 5)(x — 1)$
$frac{x^2 — 6x + 5}{x — 5} = frac{(x — 5)(x — 1)}{x — 5} = x — 1$
Ответ: x − 12) $frac{2x + 12}{x^2 + 3x — 18}$
$x^2 + 3x — 18 = 0$
$D = b^2 — 4ac = 3^2 — 4 * 1 * (-18) = 9 + 72 = 81 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-3 + sqrt{81}}{2 * 1} = frac{-3 + 9}{2} = frac{6}{2} = 3$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-3 — sqrt{81}}{2 * 1} = frac{-3 — 9}{2} = frac{-12}{2} = -6$
$x^2 + 3x — 18 = (x — 3)(x — (-6)) = (x — 3)(x + 6)$
$frac{2x + 12}{x^2 + 3x — 18} = frac{2(x + 6)}{(x — 3)(x + 6)} = frac{2}{x — 3}$
Ответ: $frac{2}{x — 3}$3) $frac{x^2 + 9x + 14}{x^2 + 7x}$
$x^2 + 9x + 14 = 0$
$D = b^2 — 4ac = 9^2 — 4 * 1 * 14 = 81 + 56 = 25 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-9 + sqrt{25}}{2 * 1} = frac{-9 + 5}{2} = frac{-4}{2} = -2$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-9 — sqrt{25}}{2 * 1} = frac{-9 — 5}{2} = frac{-14}{2} = -7$
$x^2 + 9x + 14 = (x — (-2))(x — (-7)) = (x + 2)(x + 7)$
$frac{x^2 + 9x + 14}{x^2 + 7x} = frac{(x + 2)(x + 7)}{x(x + 7)} = frac{x + 2}{x}$
Ответ: $frac{x + 2}{x}$
757. Сократите дробь:
1) $frac{4a^2 — 9}{2a^2 — 9a — 18}$;
2) $frac{2b^2 — 7b + 3}{4b^2 — 4b + 1}$;
3) $frac{c^2 — 5c — 6}{c^2 — 8c + 12}$;
4) $frac{m^3 — 1}{m^2 + 9m — 10}$;
5) $frac{x^2 — 16}{32 — 4x — x^2}$;
6) $frac{4n^2 — 9n + 2}{2 + 9n — 5n^2}$.
Решение:
1) $frac{4a^2 — 9}{2a^2 — 9a — 18}$
$2a^2 — 9a — 18 = 0$
$D = b^2 — 4ac = (-9)^2 — 4 * 2 * (-18) = 81 + 144 = 251 > 0$
$a_1 = frac{-b + sqrt{D}}{2a} = frac{9 + sqrt{251}}{2 * 2} = frac{9 + 15}{4} = frac{24}{4} = 6$
$a_2 = frac{-b — sqrt{D}}{2a} = frac{9 — sqrt{251}}{2 * 2} = frac{9 — 15}{4} = frac{-6}{4} = -1,5$
$2a^2 — 9a — 18 = 2(a — 6)(a — (-1,5)) = (a — 6)(2a + 3)$
$frac{4a^2 — 9}{2a^2 — 9a — 18} = frac{(2a — 3)(2a + 3)}{(a — 6)(2a + 3)} = frac{2a — 3}{a — 6}$
Ответ: $frac{2a — 3}{a — 6}$2) $frac{2b^2 — 7b + 3}{4b^2 — 4b + 1}$
$2b^2 — 7b + 3 = 0$
$D = b^2 — 4ac = (-7)^2 — 4 * 2 * 3 = 49 + 24 = 25 > 0$
$b_1 = frac{-b + sqrt{D}}{2a} = frac{7 + sqrt{25}}{2 * 2} = frac{7 + 5}{4} = frac{12}{4} = 3$
$b_2 = frac{-b — sqrt{D}}{2a} = frac{7 — sqrt{25}}{2 * 2} = frac{7 — 5}{4} = frac{2}{4} = 0,5$
$2b^2 — 7b + 3 = 2(b — 3)(b — 0,5) = (b — 3)(2b — 1)$
$frac{2b^2 — 7b + 3}{4b^2 — 4b + 1} = frac{(b — 3)(2b — 1)}{(2b — 1)^2} = frac{b — 3}{2b — 1}$
Ответ: $frac{b — 3}{2b — 1}$3) $frac{c^2 — 5c — 6}{c^2 — 8c + 12}$
$c^2 — 5c — 6 = 0$
$D = b^2 — 4ac = (-5)^2 — 4 * 1 * (-6) = 25 + 24 = 49 > 0$
$c_1 = frac{-b + sqrt{D}}{2a} = frac{5 + sqrt{49}}{2 * 1} = frac{5 + 7}{2} = frac{12}{2} = 6$
$c_2 = frac{-b — sqrt{D}}{2a} = frac{5 — sqrt{49}}{2 * 1} = frac{5 — 7}{2} = frac{-2}{2} = -1$
$c^2 — 5c — 6 = (c — 6)(c — (-1)) = (c — 6)(c + 1)$
$c^2 — 8c + 12 = 0$
$D = b^2 — 4ac = (-8)^2 — 4 * 1 * 12 = 64 + 48 = 16 > 0$
$c_1 = frac{-b + sqrt{D}}{2a} = frac{8 + sqrt{16}}{2 * 1} = frac{8 + 4}{2} = frac{12}{2} = 6$
$c_2 = frac{-b — sqrt{D}}{2a} = frac{8 — sqrt{16}}{2 * 1} = frac{8 — 4}{2} = frac{4}{2} = 2$
$c^2 — 8c + 12 = (c — 6)(c — 2)$
$frac{c^2 — 5c — 6}{c^2 — 8c + 12} = frac{(c — 6)(c + 1)}{(c — 6)(c — 2)} = frac{c + 1}{c — 2}$
Ответ: $frac{c + 1}{c — 2}$4) $frac{m^3 — 1}{m^2 + 9m — 10}$
$m^2 + 9m — 10 = 0$
$D = b^2 — 4ac = 9^2 — 4 * 1 * (-10) = 81 + 40 = 121 > 0$
$m_1 = frac{-b + sqrt{D}}{2a} = frac{-9 + sqrt{121}}{2 * 1} = frac{-9 + 11}{2} = frac{2}{2} = 1$
$m_2 = frac{-b — sqrt{D}}{2a} = frac{-9 — sqrt{121}}{2 * 1} = frac{-9 — 11}{2} = frac{-20}{2} = -10$
$m^2 + 9m — 10 = (m — 1)(m — (-10)) = (m — 1)(m + 10)$
$frac{m^3 — 1}{m^2 + 9m — 10} = frac{(m — 1)(m^2 + m + 1)}{(m — 1)(m + 10)} = frac{m^2 + m + 1}{m + 10}$
Ответ: $frac{m^2 + m + 1}{m + 10}$5) $frac{x^2 — 16}{32 — 4x — x^2}$
$32 — 4x — x^2 = -x^2 — 4x + 32 = 0$
$D = b^2 — 4ac = (-4)^2 — 4 * (-1) * 32 = 16 + 128 = 144 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{4 + sqrt{144}}{2 * (-1)} = frac{4 + 12}{-2} = frac{16}{-2} = -8$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{4 — sqrt{144}}{2 * (-1)} = frac{4 — 12}{-2} = frac{-8}{-2} = 4$
$-x^2 — 4x + 32 = -(x — (-8))(x — 4) = -(x + 8)(x — 4)$
$frac{x^2 — 16}{32 — 4x — x^2} = -frac{(x — 4)(x + 4)}{(x + 8)(x — 4)} = -frac{x + 4}{x + 8}$
Ответ: $-frac{x + 4}{x + 8}$6) $frac{4n^2 — 9n + 2}{2 + 9n — 5n^2}$
$4n^2 — 9n + 2 = 0$
$D = b^2 — 4ac = (-9)^2 — 4 * 4 * 2 = 81 — 32 = 49 > 0$
$n_1 = frac{-b + sqrt{D}}{2a} = frac{9 + sqrt{49}}{2 * 4} = frac{9 + 7}{8} = frac{16}{8} = 2$
$n_2 = frac{-b — sqrt{D}}{2a} = frac{9 — sqrt{49}}{2 * 4} = frac{9 — 7}{8} = frac{2}{8} = frac{1}{4}$
$4n^2 — 9n + 2 = 4(n — 2)(n — frac{1}{4}) = (n — 2)(4n — 1)$
$2 + 9n — 5n^2 = -5n^2 + 9n + 2 = 0$
$D = b^2 — 4ac = 9^2 — 4 * (-5) * 2 = 81 + 40 = 121 > 0$
$n_1 = frac{-b + sqrt{D}}{2a} = frac{-9 + sqrt{121}}{2 * (-5)} = frac{-9 + 11}{-10} = frac{2}{-10} = -0,2$
$n_2 = frac{-b — sqrt{D}}{2a} = frac{-9 — sqrt{121}}{2 * (-5)} = frac{-9 — 11}{-10} = frac{-20}{-10} = 2$
$-5n^2 + 9n + 2 = -5(n — (-0,2))(n — 2) = -5(n + 0,2)(n — 2) = -(5n + 1)(n — 2)$
$frac{4n^2 — 9n + 2}{2 + 9n — 5n^2} = frac{(n — 2)(4n — 1)}{-(5n + 1)(n — 2)} = -frac{4n — 1}{5n + 1} = frac{1 — 4n}{5n + 1}$
Ответ: $frac{1 — 4n}{5n + 1}$
758. Сократите дробь:
1) $frac{4x^2 + x — 3}{x^2 — 1}$;
2) $frac{2y^2 + 3y — 5}{y^2 — 2y + 1}$;
3) $frac{a^2 + 5a + 4}{a^2 — a — 20}$;
4) $frac{3 + 20b — 7b^2}{7b^2 — 6b — 1}$.
Решение:
1) $frac{4x^2 + x — 3}{x^2 — 1}$
$4x^2 + x — 3 = 0$
$D = b^2 — 4ac = 1^2 — 4 * 4 * (-3) = 1 + 48 = 49 > 0$
$x_1 = frac{-b + sqrt{D}}{2a} = frac{-1 + sqrt{49}}{2 * 4} = frac{-1 + 7}{8} = frac{6}{8} =frac{3}{4}$
$x_2 = frac{-b — sqrt{D}}{2a} = frac{-1 — sqrt{49}}{2 * 4} = frac{-1 — 7}{8} = frac{-8}{8} = -1$
$4x^2 + x — 3 = 4(x — frac{3}{4})(x — (-1)) = (4x — 3)(x + 1)$
$frac{4x^2 + x — 3}{x^2 — 1} = frac{(4x — 3)(x + 1)}{(x — 1)(x + 1)} = frac{4x — 3}{x — 1}$
Ответ: $frac{4x — 3}{x — 1}$2) $frac{2y^2 + 3y — 5}{y^2 — 2y + 1}$
$2y^2 + 3y — 5 = 0$
$D = b^2 — 4ac = 3^2 — 4 * 2 * (-5) = 9 + 40 = 49 > 0$
$y_1 = frac{-b + sqrt{D}}{2a} = frac{-3 + sqrt{49}}{2 * 2} = frac{-3 + 7}{4} = frac{4}{4} = 1$
$y_2 = frac{-b — sqrt{D}}{2a} = frac{-3 — sqrt{49}}{2 * 2} = frac{-3 — 7}{4} = frac{-10}{4} = -2,5$
$2y^2 + 3y — 5 = 2(y — 1)(y — (-2,5)) = 2(y — 1)(y + 2,5) = (y — 1)(2y + 5)$
$frac{2y^2 + 3y — 5}{y^2 — 2y + 1} = frac{(y — 1)(2y + 5)}{(y — 1)^2} = frac{2y + 5}{y — 1}$
Ответ: $frac{2y + 5}{y — 1}$3) $frac{a^2 + 5a + 4}{a^2 — a — 20}$
$a^2 + 5a + 4 = 0$
$D = b^2 — 4ac = 5^2 — 4 * 1 * 4 = 25 — 16 = 9 > 0$
$a_1 = frac{-b + sqrt{D}}{2a} = frac{-5 + sqrt{9}}{2 * 1} = frac{-5 + 3}{2} = frac{-2}{2} = -1$
$a_2 = frac{-b — sqrt{D}}{2a} = frac{-5 — sqrt{9}}{2 * 1} = frac{-5 — 3}{2} = frac{-8}{2} = -4$
$a^2 + 5a + 4 = (a — (-1))(a — (-4)) = (a + 1)(a + 4)$
$a^2 — a — 20 = 0$
$D = b^2 — 4ac = (-1)^2 — 4 * 1 * (-20) = 1 + 80 = 81 > 0$
$a_1 = frac{-b + sqrt{D}}{2a} = frac{1 + sqrt{81}}{2 * 1} = frac{1 + 9}{2} = frac{10}{2} = 5$
$a_2 = frac{-b — sqrt{D}}{2a} = frac{1 — sqrt{81}}{2 * 1} = frac{1 — 9}{2} = frac{-8}{2} = -4$
$a^2 — a — 20 = (a — 5)(a — (-4)) = (a — 5)(a + 4)$
$frac{a^2 + 5a + 4}{a^2 — a — 20} = frac{(a + 1)(a + 4)}{(a — 5)(a + 4)} = frac{a + 1}{a — 5}$
Ответ: $frac{a + 1}{a — 5}$4) $frac{3 + 20b — 7b^2}{7b^2 — 6b — 1}$
$3 + 20b — 7b^2 = -7b^2 + 20b + 3 = 0$
$D = b^2 — 4ac = 20^2 — 4 * (-7) * 3 = 400 + 84 = 484 > 0$
$b_1 = frac{-b + sqrt{D}}{2a} = frac{-20 + sqrt{484}}{2 * (-7)} = frac{-20 + 22}{-14} = frac{2}{-14} = -frac{1}{7}$
$b_2 = frac{-b — sqrt{D}}{2a} = frac{-20 — sqrt{484}}{2 * (-7)} = frac{-20 — 22}{-14} = frac{-42}{-14} = 3$
$-7b^2 + 20b + 3 = -7(b — (-frac{1}{7}))(b — 3) = -7(b + frac{1}{7})(b — 3) = (7b + 1)(3 — b)$
$7b^2 — 6b — 1 = 0$
$D = b^2 — 4ac = (-6)^2 — 4 * 7 * (-1) = 36 + 28 = 64 > 0$
$b_1 = frac{-b + sqrt{D}}{2a} = frac{6 + sqrt{64}}{2 * 7} = frac{6 + 8}{14} = frac{14}{14} = 1$
$b_2 = frac{-b — sqrt{D}}{2a} = frac{6 — sqrt{64}}{2 * 7} = frac{6 — 8}{14} = frac{-2}{14} = -frac{1}{7}$
$7b^2 — 6b — 1 = 7(b — 1)(b — (-frac{1}{7})) = 7(b — 1)(b + frac{1}{7}) = (b — 1)(7b + 1)$
$frac{3 + 20b — 7b^2}{7b^2 — 6b — 1} = frac{(7b + 1)(3 — b)}{(b — 1)(7b + 1)} = frac{3 — b}{b — 1}$
Ответ: $frac{3 — b}{b — 1}$
759. При каком значении b разложение на линейные множители трехчлена:
1) $2x^2 — 5x + b$ содержит множитель (x − 3);
2) $-4x^2 + bx + 2$ содержит множитель (x + 1);
3) $3x^2 — 4x + b$ содержит множитель (3x − 2)?
Решение:
1) $2x^2 — 5x + b = 2(x — 3)(x — x_2)$
$x_1 = 3$
$x_1 + x_2 = -frac{b}{a} = -frac{-5}{2} = 2,5$
$3 + x_2 = 2,5$
$x_2 = 2,5 — 3$
$x_2 = -0,5$
$x_1x_2 = frac{c}{a} = frac{b}{a} = frac{b}{2}$
$frac{b}{2} = 3 * (-0,5)$
$frac{b}{2} = -1,5$
b = −1,5 * 2
b = −3
Ответ: при b = −32) $-4x^2 + bx + 2 = -4(x + 1)(x — x_2)$
$x_1 = -1$
$x_1x_2 = frac{c}{a} = frac{2}{-4} = -0,5$
$-1 * x_2 = -0,5$
$x_2 = 0,5$
$x_1 + x_2 = -frac{b}{a} = -frac{b}{-4} = frac{b}{4}$
$frac{b}{4} = -1 + 0,5$
$frac{b}{4} = -0,5$
b = −0,5 * 4
b = −2
Ответ: при b = −23) $3x^2 — 4x + b = 3(x — x_1)(x — x_2)$
$3x — 3x_1 = 3x — 2$
$3x — 3x_1 — 3x = -2$
$-3x_1 = -2$
$x_1 = frac{2}{3}$
$x_1 + x_2 = -frac{b}{a} = -frac{-4}{3} = frac{4}{3}$
$frac{2}{3} + x_2 = frac{4}{3}$
$x_2 = frac{4}{3} — frac{2}{3}$
$x_2 = frac{2}{3}$
$x_1x_2 = frac{c}{a} = frac{b}{3}$
$frac{b}{3} = frac{2}{3} * frac{2}{3}$
$frac{b}{3} = frac{4}{9}$ |* 9
3b = 4
$b = frac{4}{3} = 1frac{1}{3}$
Ответ: при $b = 1frac{1}{3}$