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Решаем квадратное уравнение, D =49, корни +2 и -1/3.
<span>разлагается по формуле ax </span>2<span> <span>+ bx+ c = a </span></span><span>(<span> x – x</span></span>1 <span>) (<span> x – x</span></span>2 <span>)<span> .
</span></span><span>3x^2-5x-2 = 3(х +1/3) (х-2).</span>
1)
а) ![.....=-\frac{7a}{4b^{3}c^{4}}](https://tex.z-dn.net/?f=.....%3D-%5Cfrac%7B7a%7D%7B4b%5E%7B3%7Dc%5E%7B4%7D%7D)
б) ![.....=\frac{54x^{7}}{18x^{4}y^{16}} =\frac{3x^{3}}{y^{16}}](https://tex.z-dn.net/?f=.....%3D%5Cfrac%7B54x%5E%7B7%7D%7D%7B18x%5E%7B4%7Dy%5E%7B16%7D%7D+%3D%5Cfrac%7B3x%5E%7B3%7D%7D%7By%5E%7B16%7D%7D)
в) ![......=\frac{4(x+t)}{t} *\frac{3t^{2}}{(x-t)(x+t)} =\frac{12t}{x-t}](https://tex.z-dn.net/?f=......%3D%5Cfrac%7B4%28x%2Bt%29%7D%7Bt%7D+%2A%5Cfrac%7B3t%5E%7B2%7D%7D%7B%28x-t%29%28x%2Bt%29%7D+%3D%5Cfrac%7B12t%7D%7Bx-t%7D)
г) ![......=\frac{6(x-5)}{x+8} *\frac{2(x+8)}{(x+5)(x-5)} =\frac{12}{x-5}](https://tex.z-dn.net/?f=......%3D%5Cfrac%7B6%28x-5%29%7D%7Bx%2B8%7D+%2A%5Cfrac%7B2%28x%2B8%29%7D%7B%28x%2B5%29%28x-5%29%7D+%3D%5Cfrac%7B12%7D%7Bx-5%7D)
2)
a) ![.....=\frac{2a}{a-2} +\frac{a+7}{4(2-a)} *\frac{32}{a(7+a)} =\frac{2a}{a-2} +\frac{8}{a(2-a)}= \frac{2a}{a-2} -\frac{8}{a(a-2)} =\frac{2a^{2}-8}{a(a-2)} = \\ \\ \frac{2(a^{2}-4)}{a(a-2)} =\frac{2(a-2)(a+2)}{a(a-2)} =\frac{2(a+2)}{a}](https://tex.z-dn.net/?f=.....%3D%5Cfrac%7B2a%7D%7Ba-2%7D+%2B%5Cfrac%7Ba%2B7%7D%7B4%282-a%29%7D+%2A%5Cfrac%7B32%7D%7Ba%287%2Ba%29%7D+%3D%5Cfrac%7B2a%7D%7Ba-2%7D+%2B%5Cfrac%7B8%7D%7Ba%282-a%29%7D%3D+%5Cfrac%7B2a%7D%7Ba-2%7D+-%5Cfrac%7B8%7D%7Ba%28a-2%29%7D+%3D%5Cfrac%7B2a%5E%7B2%7D-8%7D%7Ba%28a-2%29%7D+%3D+%5C%5C+%5C%5C+%5Cfrac%7B2%28a%5E%7B2%7D-4%29%7D%7Ba%28a-2%29%7D+%3D%5Cfrac%7B2%28a-2%29%28a%2B2%29%7D%7Ba%28a-2%29%7D+%3D%5Cfrac%7B2%28a%2B2%29%7D%7Ba%7D)
б) ![......=\frac{(x+1)(x+1)-(x-1)(x-1)}{(x-1)(x+1)} :\frac{2x}{(1-x)(1+x)} =\frac{x^{2}+2x+1-x^{2}+2x-1}{(x-1)(x+1)} *\frac{(1-x)(1+x)}{2x} =\\ \\ \frac{4x}{(x-1)(x+1)} *\frac{(1-x)(1+x)}{2x}=\frac{2(1-x)}{x-1} =-\frac{2(x-1)}{x-1)} =-2](https://tex.z-dn.net/?f=......%3D%5Cfrac%7B%28x%2B1%29%28x%2B1%29-%28x-1%29%28x-1%29%7D%7B%28x-1%29%28x%2B1%29%7D+%3A%5Cfrac%7B2x%7D%7B%281-x%29%281%2Bx%29%7D+%3D%5Cfrac%7Bx%5E%7B2%7D%2B2x%2B1-x%5E%7B2%7D%2B2x-1%7D%7B%28x-1%29%28x%2B1%29%7D+%2A%5Cfrac%7B%281-x%29%281%2Bx%29%7D%7B2x%7D+%3D%5C%5C+%5C%5C+%5Cfrac%7B4x%7D%7B%28x-1%29%28x%2B1%29%7D+%2A%5Cfrac%7B%281-x%29%281%2Bx%29%7D%7B2x%7D%3D%5Cfrac%7B2%281-x%29%7D%7Bx-1%7D+%3D-%5Cfrac%7B2%28x-1%29%7D%7Bx-1%29%7D+%3D-2)
3) Чтобы доказать, решим первую половину.
![........=(\frac{c^{3}}{(c-4)(c-4)} -\frac{c^{2}}{c-4} ):(\frac{c^{2}}{(c-4)(c+4)} -\frac{c}{c-4} )=\\ \\ \frac{c^{3}-c^{2}(c-4)}{(c-4)(c-4)} :\frac{c^{2}-c(c+4)}{(c-4)(c+4)} =\frac{c^{3}-c^{3}+4c^{2}}{(c-4)(c-4)} *\frac{(c-4)(c+4)}{c^{2}-c^{2}-4c} =-\frac{4c^{2}(c+4)}{4c(c-4)} =\frac{c(c+4)}{4-c} =\frac{c^{2}+4c}{4-c}](https://tex.z-dn.net/?f=........%3D%28%5Cfrac%7Bc%5E%7B3%7D%7D%7B%28c-4%29%28c-4%29%7D+-%5Cfrac%7Bc%5E%7B2%7D%7D%7Bc-4%7D+%29%3A%28%5Cfrac%7Bc%5E%7B2%7D%7D%7B%28c-4%29%28c%2B4%29%7D+-%5Cfrac%7Bc%7D%7Bc-4%7D+%29%3D%5C%5C+%5C%5C+%5Cfrac%7Bc%5E%7B3%7D-c%5E%7B2%7D%28c-4%29%7D%7B%28c-4%29%28c-4%29%7D+%3A%5Cfrac%7Bc%5E%7B2%7D-c%28c%2B4%29%7D%7B%28c-4%29%28c%2B4%29%7D+%3D%5Cfrac%7Bc%5E%7B3%7D-c%5E%7B3%7D%2B4c%5E%7B2%7D%7D%7B%28c-4%29%28c-4%29%7D+%2A%5Cfrac%7B%28c-4%29%28c%2B4%29%7D%7Bc%5E%7B2%7D-c%5E%7B2%7D-4c%7D+%3D-%5Cfrac%7B4c%5E%7B2%7D%28c%2B4%29%7D%7B4c%28c-4%29%7D+%3D%5Cfrac%7Bc%28c%2B4%29%7D%7B4-c%7D+%3D%5Cfrac%7Bc%5E%7B2%7D%2B4c%7D%7B4-c%7D)
Тождество верно.
(b+c-2a)(c-b)+(c+a-2b)(a-c)-(a+b-2c)(a-b)=
=bc+c²-2ac-b²-bc+2ab+ac+a²-2ab-c²-ac+2bc-a²-ab+2ac+ab+b²-2bc=
=(bc-bc+2bc-2bc)+(ac-ac+2ac-2ac)+(ab-ab+2ab-2ab)+(a²-a²+b²-b²+c²-c²)≡0.
Не получится. Например: х куб=8 Отв: х=2, х=2, х=2