Question

\(h(x)=\sqrt{x+3}-3\)

Short answer

The solution for \(x\) in terms of \(h(x)\) is \(x = (h(x))^2 + 6h(x) + 6\).

Step-by-step solution

Isolating the square root

To isolate the square root on one side of the equation, add \(3\) to both sides of the equation. This gives \(h(x) + 3 = \sqrt{x+3}\).

Squaring both sides

To remove the square root, square both sides of the equation. This gives \((h(x) + 3)^2 = x + 3\).

Expanding and simplifying

Expand the left hand side of the equation and simplify to isolate \(x\). This results in \(x = (h(x))^2 + 6h(x) + 9 - 3\). Simplifying further, we obtain \(x = (h(x))^2 + 6h(x) + 6\).

Key concepts

Isolating the Square Root

Starting with the fundamental principle of algebra, to solve an equation containing a square root, the first step is often to isolate the square root on one side of the equation. This is crucial because we cannot directly solve for a variable if it’s inside a square root. Instead, we move all the other terms to the opposite side to set the stage for removing the square root.
For the problem \(h(x) = \sqrt{x+3} - 3\), we begin by adding 3 to both sides to isolate the square root:
\[h(x) + 3 = \sqrt{x+3}\].
It is like clearing the stage for an actor—the square root—to perform its solo act without other terms interrupting. This clear isolation allows us to then employ other methods—such as squaring—to solve for the variable in question.

Squaring Both Sides

Once we have the square root all by itself on one side of the equation, we use an 'equal' tool at our disposal: squaring both sides. Why squaring? Because it's the inverse operation of taking a square root, and doing so abolishes the root while keeping the balance of the equation intact.
For our equation \(h(x) + 3 = \sqrt{x+3}\), when we square both sides, we're effectively mirroring our action to ensure equality is maintained: \[ (h(x) + 3)^2 = (\sqrt{x+3})^2 \].
The elegance of this step lies in its simplicity—just like reflecting both sides of our face equally in a mirror, squaring keeps both sides of the equation in harmony, and we end up with an equation that no longer has a square root: \[ (h(x) + 3)^2 = x + 3 \].

Expanding and Simplifying Equations

With the square root eliminated, the next logical step is to expand and simplify the equation to solve for the variable. Expanding involves applying distributive laws, such as \( (a+b)^2 = a^2 + 2ab + b^2 \), while simplifying requires combining like terms to condense the expression.
Returning to our previous work, \( (h(x) + 3)^2 = x + 3 \), we expand the left side to give us \( h(x)^2 + 6h(x) + 9 = x + 3 \).
We then simplify by moving terms around to get \( h(x)^2 + 6h(x) + 6 = x \) after combining like terms and subtracting 3 from both sides. The move from an expanded equation to a simplified one allows us to see the solution more clearly and brings us closer to isolating the variable.
This method is akin to tidying a room—everything has its place, and once it's organized, it’s easier to find what you’re looking for, in this case, the value of \( x \).