English, asked by gauravbaghel076, 6 months ago

3
.1. Solve the equation : x°-15-126=0 by Cardan's Method.​

Answers

Answered by sumanbag0036
0

Answer:

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Answered by SrijanShrivastava
0

To Solve : For Values of x such that

 f(x) = {x}^{3}  - 15x  - 126 = 0

Here,

Degree  [f(x),x]  = 3 \\  \:  \:  \:  \:  \:  \:  \:  \:  \:  Coefficient[f(x),x] = 0

Therefore, x must be a binomial.

Calculating the nature of roots:

 \\ Discriminant[f(x) , x] =  - 27( - 126)^{2}  - 4(  { - 15})^{3}

 \triangle  =  - 415152 < 0

 \implies x  _{1} \in ℝ

Now,

Let x being a binomial.

 : =  \: x = u + v

(u + v) ^{3}  - 15(u + v) - 126 = 0

 \\  {u}^{3}  +  {v}^{3}  +3uv(u + v) - 15(u + v) - 126 = 0

 \\ ( {u}^{3}  +  {v}^{3}  - 126) + (u + v)(3uv - 15) = 0

Therefore, Simutaneously, we must conclude that

  \implies {u}^{3}   - 126 +   {v}^{3}   = 0

 \implies 3uv - 15 = 0

⇔uv = 5

Now, Multiplying First Equation by (u)³ on both sides and substituting using second equation

( {u}^{6} )  - 126( {u}^{3} ) + 125 = 0

 {u}^{3}  =  \frac{126 \pm \sqrt{(126)^{2}   - 4 (125)} }{2}

 {u}^{3}  = 63 \pm   \frac{ \sqrt{ {124}^{2} } }{2}

 {u}^{3}  = 63 \pm62

But, As u and v are indistinguishable. And, u≠v

Therefore,

  \:  \:  \:  \: {u}^{3}  = 63  +  62 = 125 \: \\  {v}^{3}  = 63 - 62 = 1 \:

And, Thus,

u = 5 \: ,  \: 5 \omega \: , \: 5 {\omega}^{2}

v = 1  \: ,  \:  \omega ^{2}   \: , \:  \omega

Therefore, All the values of x are:

 \implies x_{1} = 6

 \implies x_{2}  =   - 3 + 2 \sqrt{3} i

 \implies x_{3} =  - 3 -  2\sqrt{3} i

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