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受欢迎的代数 >

2sqrt(x-1)+2=(3x)/(x-1)

解答

2 x - 1 + 2 = \frac{3 x}{x - 1}

解答

x \approx 2.79009 \dots

求解步骤

2 x - 1 + 2 = \frac{3 x}{x - 1}

在两边乘以

x - 1

2 x - 1 (x - 1) + 2 (x - 1) = \frac{3 x}{x - 1} (x - 1)

化简

2 x - 1 (x - 1) + 2 (x - 1) = 3 x

展开

2 x - 1 (x - 1) + 2 (x - 1) : 2 x - 1 x - 2 x - 1 + 2 x - 2

2 x - 1 x - 2 x - 1 + 2 x - 2 = 3 x

因式分解

2 x - 1 x - 2 x - 1 + 2 x - 2 : 2 (x - 1) (x - 1 + 1)

2 (x - 1) (x - 1 + 1) = 3 x

两边进行平方:

4 x^{3} + 8 x^{2} x - 1 - 8 x^{2} + 4 x - 16 x x - 1 + 8 x - 1 = 9 x^{2}

4 x^{3} + 8 x^{2} x - 1 - 8 x^{2} + 4 x - 16 x x - 1 + 8 x - 1 = 9 x^{2}

将

4 x^{3}

到右边

8 x^{2} x - 1 - 8 x^{2} + 4 x - 16 x x - 1 + 8 x - 1 = 9 x^{2} - 4 x^{3}

将

8 x^{2}

到右边

8 x^{2} x - 1 + 4 x - 16 x x - 1 + 8 x - 1 = - 4 x^{3} + 17 x^{2}

将

4 x

到右边

8 x^{2} x - 1 - 16 x x - 1 + 8 x - 1 = - 4 x^{3} + 17 x^{2} - 4 x

分解

8 x^{2} x - 1 - 16 x x - 1 + 8 x - 1 : 8 x - 1 (x^{2} - 2 x + 1)

8 x - 1 (x^{2} - 2 x + 1) = - 4 x^{3} + 17 x^{2} - 4 x

两边进行平方:

64 x^{5} - 320 x^{4} + 640 x^{3} - 640 x^{2} + 320 x - 64 = 16 x^{6} - 136 x^{5} + 321 x^{4} - 136 x^{3} + 16 x^{2}

64 x^{5} - 320 x^{4} + 640 x^{3} - 640 x^{2} + 320 x - 64 = 16 x^{6} - 136 x^{5} + 321 x^{4} - 136 x^{3} + 16 x^{2}

解

64 x^{5} - 320 x^{4} + 640 x^{3} - 640 x^{2} + 320 x - 64 = 16 x^{6} - 136 x^{5} + 321 x^{4} - 136 x^{3} + 16 x^{2} : x \approx 2.79009 \dots, x \approx 8.31703 \dots

x \approx 2.79009 \dots, x \approx 8.31703 \dots

验证解:

x \approx 2.79009 \dots

真

, x \approx 8.31703 \dots

假

解是

x \approx 2.79009 \dots

作图

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x + x - 10 = 2 - x - x

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2 x^{2} + 3 x - 2 = 0

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a (x) = - x + 3