Question

Solve the equation. Check for extraneous solutions. $$-5-\sqrt{10 x-2}=5$$

Short answer

The equation has no solution.

Step-by-step solution

Isolate the square root

Re-arrange the equation to put the square root term alone on one side. This can be done by adding 5 to both sides of the equation \(-5-\sqrt{10 x-2}=5\), which gives \(\sqrt{10 x-2}=10\).

Square both sides

Square both sides of the equation to eliminate the square root. This gives, \(10 x-2= 100.\)

Solve for \(x\)

Simplify and solve for \(x\). By adding 2 to each side, we get \(10x = 102\). Dividing every side by 10 gives \(x = 10.2\).

Check for extraneous solutions

Substitute 10.2 into initial equation to confirm if it's a valid solution. This yields -5 - sqrt{(10*10.2) - 2} = 5, which simplifies to -5 - sqrt{100} = 5. This becomes -5 - 10 = 5, which simplifies further to -15 = 5, which is incorrect. Hence, 10.2 is an extraneous solution and is not a solution to the original equation. Thus, the equation has no solution.