Solve the equation
27x^4 - 195x^3 + 494x^2 - 520x + 192, the roots being in G.P. Please do not use the Factor Theorem.

We have, 

.x^4 - frac{195}{27}x^3 + frac{494}{27} - frac{520}{27}x + frac{64}{9} := :0

If the roots are

 a,b,c,dtext{, then: }abcd = frac{64}{9};text{ and };abc + abd + acd + bcd = frac{520}{27}

Let the roots be, 

 a,;ar,,ar^2,;ar^3 
$(a)(ar)(ar^2)(ar^3) = frac{64}{9}quadRightarrowquad a^4r^6 = frac{64}{9}quadRightarrowquad a^2r^3 = frac{8}{3} ..........................(1)

.$(a)(ar)(ar^2) + (a)(ar)(ar^3) + (a)(ar^2)(ar^3) + (ar)(ar^2)(ar^3) ;= ;frac{520}{27}
$a^3r^3 + a^3r^4 + a^3r^5 + a^3r^6 = frac{520}{27}
 $a^3r^3(1 + r + r^2 + r^3) = frac{520}{27}  .....................(2)
Now using equation 1 and 2 we get, 83×1+r+r2+r3=520271+r+r2+r3=6599r3+9r2+9r+9=659r3+9r2+9r-56=0Now the roots of this equation are not real so kindly recheck the question.



 

  • -4
What are you looking for?