Question

Find the non-extraneous solutions of the square root of the quantity x plus 6 minus 5 equals quantity x plus 1.

x = −2
x = −6 and x = −5
x = −6 and x = −2
x = 2

Answer 1

x = −6 and x = −7

Answer 2

The radical equation is
`√(x+6)-5=x+1`.


i) We first isolate the square root, adding 5 to both sides of the equation:


`√(x+6)=x+6.`

ii) Here let's substitute x+6 with t. Doing so we have:


`√(t)=t.`

Squaring both sides, we get:


`t=t^2`

iii) Collecting the variables on the same side, and factorizing t we have:


`t(t-1)=0`, which yields

t=0 or t=1.

Now we solve for x in x+6=t:

x+6=0 ⇒x=-6 and x+6=1⇒x=-5.


iv) Now we check these values in the original equation
`√(x+6)=x+6` :

a)
`√(-6+6)=-6+6.` ⇒ 0=0 ; Correct.


b)
`√(-5+6)=-5+6.` ⇒ 1=1 ; Correct.



Answer: x = −6 and x = −5