1. $x^3 - (sqrt2 + 3)x^2 + (3sqrt2 - 4)x + 4sqrt2 = 0$
解答:
可参考例题2、例题9. $$f(-1) = 0 Rightarrow f(x) = (x+1)left[x^2 - (4+sqrt2)x ight] = (x+1)(x-4)(x-sqrt2)$$ $$Rightarrow x_1 = -1, x_2 = 4, x_3 = sqrt2.$$
2. $(6x + 7)^2(3x + 4)(x + 1) = 6$
解答:
可参考例题4.
令 $y = 6x + 7$, $$Rightarrow y^2(y+1)(y-1) = 72 Rightarrow y^4 - y^2 - 72 = 0 Rightarrow (y^2 - 9)(y^2 + 8) = 0$$ $$Rightarrow y = pm3.$$ $$Rightarrow x_1 = -frac{2}{3}, x_2 = -frac{5}{3}.$$
3. $6x^4 + 5x^3 - 38x^2 + 5x + 6 = 0$
解答:
可参考例题8. $$6(x^2 + 1)^2 + 5x(x^2 + 1) - 50x^2 = 0$$ $$Rightarrow left[2(x^2+1) - 5x ight]left[3(x^2+1) + 10x ight] = 0$$ $$Rightarrow (2x^2 - 5x + 2)(3x^2 + 10x + 3) = 0$$ $$Rightarrow x_1 = frac{1}{2}, x_2 = 2, x_3 = -frac{1}{3}, x_4 = -3.$$
4. $(x-2)(x-1)(x+3)(x+4) = 84$
解答:
可参考例题3. $$(x^2 + 2x - 8)(x^2 + 2x - 3) = 84$$ $$Rightarrow (x^2 + 2x)^2 - 11(x^2 + 2x) - 60 = 0$$ $$Rightarrow (x^2 + 2x - 15)(x^2 + 2x + 4) = 0$$ $$Rightarrow x_1 = -5, x_2 = 3.$$
5. $x^4 + (x-4)^4 = 626$
解答:
可参考例题5.
令 $y = x-2$, $$Rightarrow (y-2)^4 + (y+2)^4 = 626 Rightarrow 2y^4 + 48y^2 - 594 = 0$$ $$Rightarrow (y^2 + 33)(y^2 - 9) = 0 Rightarrow y = pm 3.$$ $$Rightarrow x_1 = 5, x_2 = -1.$$
6. $4(2x^2 - 3x - 1)(x^2 - x + 2) - (3x^2 - 4x + 1)^2 = 0$
解答:
可参考例题6.
令 $m = 2x^2 - 3x - 1$, $n = x^2 - x + 2$, $$Rightarrow 4mn - (m + n)^2 = 0$$ $$Rightarrow m = n Rightarrow 2x^2 - 3x - 1 = x^2 - x + 2$$ $$Rightarrow x^2 - 2x - 3 = 0 Rightarrow x_1 = 3, x_2 = -1.$$
7. $dfrac{x^2 + 2x + 2}{x + 1} + dfrac{x^2 + 8x + 20}{x + 4} = dfrac{x^2 + 4x + 6}{x+2} + dfrac{x^2 + 6x + 12}{x+3}$
解答:
可参考例题10. $$frac{(x+1)^2 + 1}{x+1} + frac{(x+4)^2 + 4}{x+4} = frac{(x+2)^2 + 2}{x+2} + frac{(x+3)^2 + 3}{x+3}$$ $$Rightarrow x+1 + frac{1}{x+1} + x+4 + frac{4}{x+4} = x+2 + frac{2}{x+2} + x+3 + frac{3}{x+3}$$ $$Rightarrow frac{1}{x+1} + frac{4}{x+4} = frac{2}{x+2} + frac{3}{x+3}$$ $$Rightarrow x_1 = 0, x_2 = -frac{5}{2}.$$ 经检验 $x_1 = 0, x_2 = -dfrac{5}{2}$ 均为方程的解.
主讲教师:
赵胤, 理学硕士(数学) & 教育硕士(数学), 中国数学奥林匹克一级教练员, 高级中学数学教师资格.