Question

Prove that x=√-5x-6 has no solution.

Answer 1

See explanation below.

Explanation:

We need to prove that there are no solutions of the equation: x =
`√(-5x-6)`

Let's start trying to solve this equation:
`x = √(-5x+6) \\x^(2) =-5x + 6\\x^(2) +5x - 6 = 0`

To solve this equation, by factorizing the equation we get:

`x^(2) +5x-6 = 0\\(x +6)(x-1) = 0\\x=-6\\x=1`

• Now we're going to substitute these numbers in the original equation:

For x = -6 we have:

x =
`x=√(5x-6) \\6=√(5(-6)-6) \\6=√(-30-6) \\6= √(-36)`

But √-36 has no solution in the real numbers and therefore it cannot equal 6.

• Now the second, x = 1

`x = √(-5x-6) \\1=√(-5(1)-6)\\1=√(-5-6) \\1=√(-11) \\1\\eq √(-11)`

Since the left side is different than the right one, this is not a solution.

Therefore the equation has no solution in the real numbers