Question:
Show that any root of the equation $\displaystyle 5 + x - \sqrt{3+4x} = 0$ is also a root of the equation $\displaystyle x^2 + 6x + 22 = 0$. Hence show that the equation $\displaystyle 5 + x - \sqrt{3+4x}$ has no solutions.
Question:
Show that any root of the equation $\displaystyle 5 + x - \sqrt{3+4x} = 0$ is also a root of the equation $\displaystyle x^2 + 6x + 22 = 0$. Hence show that the equation $\displaystyle 5 + x - \sqrt{3+4x}$ has no solutions.
Hello,
It's the same as $\displaystyle 5 + x = \sqrt{3+4x}$ <=> $\displaystyle (5+x)^2=3+4x$ <=> $\displaystyle 25+10x+x^2=3+4x$ <=> $\displaystyle x^2+6x+22=0$
Since these are equivalences, the roots for one will be the roots for the other.
If you want to show that there is no solution, prove that the discriminant is negative, meaning that the term is strictly positive and is never equal to 0.