Given 4x-3/x^2=m , if -4 is one of the roots determine the value of m and other roots

Question

4X-3/x^2=m

Answer 1

4x-3/x²=m could be interpreted in two ways:

(1) 4x-(3/x²)=m,

4x³-3=mx²,

4x³-mx²-3=0.

Since x=-4 is a root (a zero) then x+4 is a factor. Synthetic division:

-4 | 4      -m          0               -3

      4     -16 64+4m | -256-16m

      4 -16-m 64+4m |        0         ⇒-259-16m=0, m=-259/16; the other factor is the quadratic:

4x²-(16-m)x+4m+64 which becomes 4x²+3x/16-¾ after substituting for m.

CHECK: (x+4)(4x²+3x/16-¾)=4x³+3x²/16-3x/4+16x²+3x/4-3=4x³+259x²/16-3=0,

4x+259/16-3/x²=0, 4x-(3/x²)=-259/16=m. So this result is confirmed.

To find other roots we need to solve the quadratic 4x²+3x/16-¾=0.

We can use the quadratic formula: 

x=(-3/16±√(9/256+12))/8=(-0.1875±3.4692)/8=0.4102 or -0.4571 (approx) for the other two roots.

(2) (4x-3)/x²=m,

4x-3=mx²,

mx²-4x+3=0. Since x=-4 is a solution, we can use synthetic division:

-4 | m   -4                  3

      m -4m    | 16m+16

      m -4m-4 | 16m+19⇒16m+19=0, m=-19/16.

Another way, using x=-4 in the equation: mx²-4x+3=0⇒16m+16=-3, m=-19/16.

CHECK: (x+4)(-19x/16+¾)=-19x²/16+3x/4-19x/4+3=-19x²16-4x+3=0, 4x-3=-19x²/16, (4x-3)/x²=-19/16, confirming the result.

The other root is x=12/19 (that is, mx-4(m+1), then substitute for m, as given by the quotient of the synthetic division).

NOTE: If m=1 then 4x-3=x², x²-4x+3=0=(x-3)(x-1). This would seem an attractive solution for m, but for the fact that x=-4 is not a root.