  Answer.AI Which statement describes if there is an extraneous solution to the equation √x-3 = x-5? A. there are no solutions to the equation, B. the extraneous solution is x = 7, C. the valid solutions are x = 7 and x = 4, or D. the extraneous solution is x = 4 Solve problem with AI  Remember that an extraneous solution of an equation, is the solution that emerges from solving the equation but is not a valid solution.

Lets solve our equation to find out what is the extraneous solution:
$$\sqrt{x-3} =x-5$$
$$(\sqrt{x-3})^2 =(x-5)^2$$
$$x-3=x^2-10x+25$$
$$x^2-11x+28=0$$
$$(x-4)(x-7)=0$$
$$x-4=0$$ and $$x-7=0$$
$$x=4$$ and $$x=7$$

So, the solutions of our equation are
$$x=4$$ and $$x=7$$. Lets replace each solution in our original equation to check if they are valid solutions:
- For $$x=7$$
$$\sqrt{x-3} =x-5$$
$$\sqrt{7-3} =7-5$$
$$\sqrt{4} =2$$
$$2=2$$
We can conclude that 7 is a valid solution of the equation.

- For $$x=4$$
$$\sqrt{x-3} =x-5$$
$$\sqrt{4-3} =4-5$$
$$\sqrt{1} =1$$
$$1 \neq 1$$
We can conclude that 4 is not a valid solution of the equation; therefore, 4 is a extraneous solution.

We can conclude that the correct answer is: D. the extraneous solution is x = 4
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