Question

An in depth explanation would be appreciate, but just the answer is fine if it is quicker.

Answer 1

Explanation:

In-Depth Explanation:

√(x²-1)²= x²-1

Here we see that the √(x²-1)² can be simplified right off the bat.

Notice the (x²-1) is squared. Notice also that this squared expression is under a square root. When a something squared is under a square root, the square gets cancelled out.

Take a look at this:

√(x²-1)(x²-1)

It's the same thing as:

√(x²-1)²

Because squaring an expression or number means multiplying the expression or number by itself.

If we are to carry out the square root with the expression √(x²-1)(x²-1), it would simplify down to (x²-1) correct? The same concept can be applied to √(x²-1)². Since the square root is being taken of the expression (x²-1)² alone simplifies it down to (x²-1). Square roots cancel the squaring of expressions and numbers.

Please notice, however, that since the expression (x²-1) is squared, the square root only takes care of cancelling out the squaring of the expression (x²-1)², not the square inside of the expression (x²-1). This is due to the set of parentheses around the expression, signifying that square is squaring the WHOLE expression, not one specific part IN the expression.

Simple Explanation:

√(x²-1)²= x²-1

Square roots cancel out the squaring of the expression (x²-1)²

(x²-1)= x²-1

x²-1= x²-1

From here, we can see that both sides of the equation are identical.

x²-1= x²-1

Add +1 on boths sides to get rid of the 1s on boths sides.

x²-1+1= x²-1+1

x² = x²

Again, we can see that x² = x², but to boil it down even more we could subtract x² from both sides.

x²-x²= x²-x²

0 = 0

Both sides of this equation are equal.

Answer 2

`\bf √((x^2-1)^2) = x^2-1\implies \sqrt[2]{(x^2-1)^2} = x^2-1\implies (x^2-1) = x^2-1 \\\\\\ 0 = 0\qquad \impliedby \begin{array}{llll} \textit{dependent consistent system}\\ \textit{infinitely many solutions} \end{array}`

whenever we end up with an expression on the left-hand-side that is equal to the equation on the right-hand-side, is a way of saying, the system of equations, contains two lines that are exactly the same thing, but one is in disguise, something like say

9x + 15 = (3x + 5)3

now, if we distribute the 3 on the right-hand-side, we'd end up with the same equaton on the left-hand-side, so both are the same.

graphing wise, a solution is where both graphs intersect, when two equations are equal they'll intersect, well, at every single point, since their graph is really just one graph pancaked on top of the other, since they touch each other at every point, infinitely many solutions.