Solve the equation after making an appropriate substitution:

Question

4x^-2 - 11x^-1 - 20=0

Determine the appropriate substitution using the new u: u=

Write the new quadratic equation using the variable u:

Solve the new quadratic equation for variable u. u =

The solution set the original equation is { }

Answer 1

We need the equation to have positive exponents, since quadratic equations have a degree of 2 and not -2.

 

 

Let  x^-1 = u

 

If we square both sides of the equated variable,

 

(x^-1)² = u²

 

x^-2 = u²

 

 

Now we can substitute the u into the equation.

 

4u² + 11u - 20 = 0

 

(4u + 5)(u - 4) = 0

 

 

Set the factors equal to zero.

 

4u + 5 = 0            and             u - 4 = 0

 

u = -5/4                 and            u = 4

 

 

 

Now we take the solution of x.  Remember that

 

x^-1 = u

 

(x^-1)^-1 = u^-1

 

x = 1 / u

 

 

x = 1 / (-5 / 4)                       and                  x = 1 / 4

 

x = -4/5                                                        x = 1/4

 

 

 

 

 

Answer 2

 

4x^(-2)-11x^(-1)-20=0

let u=√x^(-2) so u^2 would equal x^(-2)

√x^(-2)=√(1/x^2)=1/x

so u=1/x and u^2=1/x^2=x^(-2)

substitute

4u^2-11u-20=0

4*20=2*2*2*2*5=(2*2*2*2)*5 and 16-5=11

factor

(u-4)(4u+5)=0

u-4=0

u=4

4u+5=0

4u=-5

u=-5/4

we set u=1/x

4=1/x so 4x=1 and x=1/4 (one answer)

u=1/x

-5/4=1/x

cross multiply

4=-5x

x=-4/5 (second answer)

 

{1/4, -4/5}