Question

How do you solve `sqrt(3x^2-11)=x+1` and identify any restrictions?

Answer 1

`x=3`

Explanation:

`color(blue)"square both sides"`

`rArr(sqrt(3x^2-11))^2=(x+1)^2`

`rArr3x^2-11=x^2+2x+1`

`rArr2x^2-2x-12=0larrcolor(blue)"in standard form"`

`rArr2(x^2-x-6)=0`

`rArr2(x-3)(x+2)=0`

`"equate each factor to zero and solve for x"`

`x-3=0rArrx=3`

`x+2=0rArrx=-2`

`color(blue)"As a check"`

Substitute these values into the equation and if both sides are equal then they are the solutions.

`sqrt(27-11)=sqrt16=4" and "3+1=4`

`sqrt(12-11)=1" and "-2+1=-1`

`"the solution is "x=3`

`x=-2" is an extraneous solution"`