求解 5/6x+18+x+9/2x

$$\frac { 5 } { 6 x + 18 } + \frac { x + 9 } { 2 x }$$

求值

$$\frac{3x^{2}+41x+81}{6x\left(x+3\right)}$$

简短求解步骤

因式分解 6x+18。

$$\frac{5}{6\left(x+3\right)}+\frac{x+9}{2x}$$

若要对表达式执行加法或减法运算，请重写该表达式，使其分母相同。 6\left(x+3\right) 和 2x 的最小公倍数是 6x\left(x+3\right)。 求 \frac{5}{6\left(x+3\right)} 与 \frac{x}{x} 的乘积。 求 \frac{x+9}{2x} 与 \frac{3\left(x+3\right)}{3\left(x+3\right)} 的乘积。

$$\frac{5x}{6x\left(x+3\right)}+\frac{\left(x+9\right)\times 3\left(x+3\right)}{6x\left(x+3\right)}$$

由于 \frac{5x}{6x\left(x+3\right)} 和 \frac{\left(x+9\right)\times 3\left(x+3\right)}{6x\left(x+3\right)} 具有相同的分母，可通过分子相加来求和。

$$\frac{5x+\left(x+9\right)\times 3\left(x+3\right)}{6x\left(x+3\right)}$$

完成 5x+\left(x+9\right)\times 3\left(x+3\right) 中的乘法运算。

$$\frac{5x+3x^{2}+9x+27x+81}{6x\left(x+3\right)}$$

合并 5x+3x^{2}+9x+27x+81 中的项。

$$\frac{41x+3x^{2}+81}{6x\left(x+3\right)}$$

将 \frac{41x+3x^{2}+81}{6x\left(x+3\right)} 中尚未因式分解的表达式分解因式。

$$\frac{3\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{6x\left(x+3\right)}$$

消去分子和分母中的 3。

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x\left(x+3\right)}$$

展开 2x\left(x+3\right)。

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

要查找 -\frac{1}{6}\sqrt{709}-\frac{41}{6} 的相反数，请查找每一项的相反数。

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}\right)-\left(-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

-\frac{1}{6}\sqrt{709} 的相反数是 \frac{1}{6}\sqrt{709}。

$$\frac{\left(x+\frac{1}{6}\sqrt{709}-\left(-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

-\frac{41}{6} 的相反数是 \frac{41}{6}。

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

要查找 \frac{1}{6}\sqrt{709}-\frac{41}{6} 的相反数，请查找每一项的相反数。

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\frac{1}{6}\sqrt{709}-\left(-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

-\frac{41}{6} 的相反数是 \frac{41}{6}。

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)}{2x^{2}+6x}$$

应用分配律，将 x+\frac{1}{6}\sqrt{709}+\frac{41}{6} 的每一项和 x-\frac{1}{6}\sqrt{709}+\frac{41}{6} 的每一项分别相乘。

$$\frac{x^{2}+x\left(-\frac{1}{6}\right)\sqrt{709}+x\times \frac{41}{6}+\frac{1}{6}\sqrt{709}x+\frac{1}{6}\sqrt{709}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

将 \sqrt{709} 与 \sqrt{709} 相乘，得到 709。

$$\frac{x^{2}+x\left(-\frac{1}{6}\right)\sqrt{709}+x\times \frac{41}{6}+\frac{1}{6}\sqrt{709}x+\frac{1}{6}\times 709\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

合并 x\left(-\frac{1}{6}\right)\sqrt{709} 和 \frac{1}{6}\sqrt{709}x，得到 0。

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{1}{6}\times 709\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

将 \frac{1}{6} 与 709 相乘，得到 \frac{709}{6}。

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{709}{6}\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

\frac{709}{6} 乘以 -\frac{1}{6} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{709\left(-1\right)}{6\times 6}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

以分数形式 \frac{709\left(-1\right)}{6\times 6} 进行乘法运算。

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{-709}{36}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

可通过提取负号，将分数 \frac{-709}{36} 重写为 -\frac{709}{36}。

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

\frac{1}{6} 乘以 \frac{41}{6} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{1\times 41}{6\times 6}\sqrt{709}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

以分数形式 \frac{1\times 41}{6\times 6} 进行乘法运算。

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

合并 x\times \frac{41}{6} 和 \frac{41}{6}x，得到 \frac{41}{3}x。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

\frac{41}{6} 乘以 -\frac{1}{6} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41\left(-1\right)}{6\times 6}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

以分数形式 \frac{41\left(-1\right)}{6\times 6} 进行乘法运算。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{-41}{36}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

可通过提取负号，将分数 \frac{-41}{36} 重写为 -\frac{41}{36}。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}-\frac{41}{36}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

合并 \frac{41}{36}\sqrt{709} 和 -\frac{41}{36}\sqrt{709}，得到 0。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

\frac{41}{6} 乘以 \frac{41}{6} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41\times 41}{6\times 6}}{2x^{2}+6x}$$

以分数形式 \frac{41\times 41}{6\times 6} 进行乘法运算。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{1681}{36}}{2x^{2}+6x}$$

由于 -\frac{709}{36} 和 \frac{1681}{36} 具有相同的分母，可通过分子相加来求和。

$$\frac{x^{2}+\frac{41}{3}x+\frac{-709+1681}{36}}{2x^{2}+6x}$$

-709 与 1681 相加，得到 972。

$$\frac{x^{2}+\frac{41}{3}x+\frac{972}{36}}{2x^{2}+6x}$$

972 除以 36 得 27。

$$\frac{x^{2}+\frac{41}{3}x+27}{2x^{2}+6x}$$

展开

$$\frac{3x^{2}+41x+81}{6x\left(x+3\right)}$$

简短求解步骤

因式分解 6x+18。

$$\frac{5}{6\left(x+3\right)}+\frac{x+9}{2x}$$

若要对表达式执行加法或减法运算，请重写该表达式，使其分母相同。 6\left(x+3\right) 和 2x 的最小公倍数是 6x\left(x+3\right)。 求 \frac{5}{6\left(x+3\right)} 与 \frac{x}{x} 的乘积。 求 \frac{x+9}{2x} 与 \frac{3\left(x+3\right)}{3\left(x+3\right)} 的乘积。

$$\frac{5x}{6x\left(x+3\right)}+\frac{\left(x+9\right)\times 3\left(x+3\right)}{6x\left(x+3\right)}$$

由于 \frac{5x}{6x\left(x+3\right)} 和 \frac{\left(x+9\right)\times 3\left(x+3\right)}{6x\left(x+3\right)} 具有相同的分母，可通过分子相加来求和。

$$\frac{5x+\left(x+9\right)\times 3\left(x+3\right)}{6x\left(x+3\right)}$$

完成 5x+\left(x+9\right)\times 3\left(x+3\right) 中的乘法运算。

$$\frac{5x+3x^{2}+9x+27x+81}{6x\left(x+3\right)}$$

合并 5x+3x^{2}+9x+27x+81 中的项。

$$\frac{41x+3x^{2}+81}{6x\left(x+3\right)}$$

将 \frac{41x+3x^{2}+81}{6x\left(x+3\right)} 中尚未因式分解的表达式分解因式。

$$\frac{3\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{6x\left(x+3\right)}$$

消去分子和分母中的 3。

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x\left(x+3\right)}$$

展开 2x\left(x+3\right)。

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

要查找 -\frac{1}{6}\sqrt{709}-\frac{41}{6} 的相反数，请查找每一项的相反数。

$$\frac{\left(x-\left(-\frac{1}{6}\sqrt{709}\right)-\left(-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

-\frac{1}{6}\sqrt{709} 的相反数是 \frac{1}{6}\sqrt{709}。

$$\frac{\left(x+\frac{1}{6}\sqrt{709}-\left(-\frac{41}{6}\right)\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

-\frac{41}{6} 的相反数是 \frac{41}{6}。

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\left(\frac{1}{6}\sqrt{709}-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

要查找 \frac{1}{6}\sqrt{709}-\frac{41}{6} 的相反数，请查找每一项的相反数。

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\frac{1}{6}\sqrt{709}-\left(-\frac{41}{6}\right)\right)}{2x^{2}+6x}$$

-\frac{41}{6} 的相反数是 \frac{41}{6}。

$$\frac{\left(x+\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)\left(x-\frac{1}{6}\sqrt{709}+\frac{41}{6}\right)}{2x^{2}+6x}$$

应用分配律，将 x+\frac{1}{6}\sqrt{709}+\frac{41}{6} 的每一项和 x-\frac{1}{6}\sqrt{709}+\frac{41}{6} 的每一项分别相乘。

$$\frac{x^{2}+x\left(-\frac{1}{6}\right)\sqrt{709}+x\times \frac{41}{6}+\frac{1}{6}\sqrt{709}x+\frac{1}{6}\sqrt{709}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

将 \sqrt{709} 与 \sqrt{709} 相乘，得到 709。

$$\frac{x^{2}+x\left(-\frac{1}{6}\right)\sqrt{709}+x\times \frac{41}{6}+\frac{1}{6}\sqrt{709}x+\frac{1}{6}\times 709\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

合并 x\left(-\frac{1}{6}\right)\sqrt{709} 和 \frac{1}{6}\sqrt{709}x，得到 0。

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{1}{6}\times 709\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

将 \frac{1}{6} 与 709 相乘，得到 \frac{709}{6}。

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{709}{6}\left(-\frac{1}{6}\right)+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

\frac{709}{6} 乘以 -\frac{1}{6} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{709\left(-1\right)}{6\times 6}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

以分数形式 \frac{709\left(-1\right)}{6\times 6} 进行乘法运算。

$$\frac{x^{2}+x\times \frac{41}{6}+\frac{-709}{36}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

可通过提取负号，将分数 \frac{-709}{36} 重写为 -\frac{709}{36}。

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{1}{6}\sqrt{709}\times \frac{41}{6}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

\frac{1}{6} 乘以 \frac{41}{6} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{1\times 41}{6\times 6}\sqrt{709}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

以分数形式 \frac{1\times 41}{6\times 6} 进行乘法运算。

$$\frac{x^{2}+x\times \frac{41}{6}-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41}{6}x+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

合并 x\times \frac{41}{6} 和 \frac{41}{6}x，得到 \frac{41}{3}x。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41}{6}\left(-\frac{1}{6}\right)\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

\frac{41}{6} 乘以 -\frac{1}{6} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{41\left(-1\right)}{6\times 6}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

以分数形式 \frac{41\left(-1\right)}{6\times 6} 进行乘法运算。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}+\frac{-41}{36}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

可通过提取负号，将分数 \frac{-41}{36} 重写为 -\frac{41}{36}。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{36}\sqrt{709}-\frac{41}{36}\sqrt{709}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

合并 \frac{41}{36}\sqrt{709} 和 -\frac{41}{36}\sqrt{709}，得到 0。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41}{6}\times \frac{41}{6}}{2x^{2}+6x}$$

\frac{41}{6} 乘以 \frac{41}{6} 的计算方法是，将两数分子与分子相乘得到分子，分母与分母相乘得到分母。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{41\times 41}{6\times 6}}{2x^{2}+6x}$$

以分数形式 \frac{41\times 41}{6\times 6} 进行乘法运算。

$$\frac{x^{2}+\frac{41}{3}x-\frac{709}{36}+\frac{1681}{36}}{2x^{2}+6x}$$

由于 -\frac{709}{36} 和 \frac{1681}{36} 具有相同的分母，可通过分子相加来求和。

$$\frac{x^{2}+\frac{41}{3}x+\frac{-709+1681}{36}}{2x^{2}+6x}$$

-709 与 1681 相加，得到 972。

$$\frac{x^{2}+\frac{41}{3}x+\frac{972}{36}}{2x^{2}+6x}$$

972 除以 36 得 27。

$$\frac{x^{2}+\frac{41}{3}x+27}{2x^{2}+6x}$$