Rearrange formulas to isolate a chosen quantity, including formulas with quadratic terms.
Rearrange a formula with a squared expression to solve for the variable inside the square
Problem
Solve \(y=a(x-h)^{2}+k\) for \(x\) over the real numbers. Isolate the squared expression, retain both square-root branches, and state the restrictions on \(a\) and the radicand.
Big Picture
What this problem is really about
The variable is buried inside a shifted square, so we isolate that entire squared expression before taking a root. We’ll remove the outside shift, divide by the nonzero multiplier, retain both square-root branches, and state that the resulting radicand must be nonnegative for real solutions before undoing the inner shift.
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