Question

solve the equation: x3-15-126=0 by cardon's method​

Answer 1

Given:

An equation :-   $x^{3} - 15x - 126 = 0$

To Solve:

Solve the above equation by Cardan's method.

Solution:

$x^{3} - 15x - 126 = 0$

Let, $x = u + v$, substitute value of x in above equation,

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

Now, put  $(3uv-15)=0$

the above equation become-

$u^{3}+v^{3}=126$

$u^{3} v^{3}=74088$

Since, this equation specifies both the sum and product of u^3 and v^3 , it enables us to determine a quadratic equation whose roots are u^3 and v^3.

Thus the equation is -

$t^{2} -15t+74088=0$

with solutions

$u^{3}=-\frac{-15}{2} +\sqrt{\frac{-15^{2} }{4}+\frac{1}{27} } ;\\ v^{3}=-\frac{-15}{2} -\sqrt{\frac{-15^{2} }{4}+\frac{1}{27} }$