Answer:
D. There are no extraneous solutions to the equation.
Step-by-step explanation:
We have been given an equation [tex]\sqrt{8x+9}=x+2[/tex]. We are asked to find extraneous solution to the radical equation.
First of all, we will square both sides of our given equation as:
[tex](\sqrt{8x+9})^2=(x+2)^2[/tex]
[tex]8x+9=x^2+4x+4[/tex]
Upon switching the sides, we will get:
[tex]x^2+4x+4=8x+9[/tex]
[tex]x^2+4x-8x+4=8x-8x+9[/tex]
[tex]x^2-4x+4=9[/tex]
[tex]x^2-4x+4-9=9-9[/tex]
[tex]x^2-4x-5=0[/tex]
Upon splitting the middle term:
[tex]x^2-5x+x-5=0[/tex]
[tex]x(x-5)+1(x-5)=0[/tex]
[tex](x-5)(x+1)=0[/tex]
Using zero product property, we will get:
[tex](x-5)=0\text{ (or) }(x+1)=0[/tex]
[tex]x=5 \text{ (or) }x=-1[/tex]
Now, we will check both solutions to find any extraneous solution as:
[tex]\sqrt{8x+9}=x+2[/tex]
[tex]\sqrt{8(-1)+9}=-1+2[/tex]
[tex]\sqrt{-8+9}=1[/tex]
[tex]\sqrt{1}=1[/tex]
[tex]1=1[/tex] True.
[tex]\sqrt{8x+9}=x+2[/tex]
[tex]\sqrt{8(5)+9}=5+2[/tex]
[tex]\sqrt{40+9}=7[/tex]
[tex]\sqrt{49}=7[/tex]
[tex]7=7[/tex] True.
Therefore, there is no extraneous solution to our given equation and option D is the correct choice.
