Solve: $(x+7)^2-7=25$
The goal is to solve the quadratic equation \((x+7)^2 - 7 = 25\) for \(x\). We observe that this equation is structured such that the variable \(x\) is contained within a squared binomial. The strategy involves isolating the binomial term, then using square roots to solve for \(x\).
To isolate the squared term \((x+7)^2\), we add 7 to both sides of the equation:
\[(x+7)^2 - 7 + 7 = 25 + 7\]
\[(x+7)^2 = 32\]
Next, we take the square root of both sides, keeping in mind that the square root of a positive constant yields both a positive and a negative solution:
\[x + 7 = \pm\sqrt{32}\]
We can simplify \(\sqrt{32}\) by identifying the largest perfect square factor of 32, which is 16:
\[\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}\]
Substituting this back, we have:
\[x + 7 = \pm4\sqrt{2}\]
Finally, we isolate \(x\) by subtracting 7 from both sides:
\[x = -7 \pm 4\sqrt{2}\]
Comparing this result to the given options, we find that the correct choice is \(x = -7 \pm 4\sqrt{2}\).
Solving quadratic equations like this is essential for calculating dimensions, such as finding the side length of a square when the area is given as a function, or determining the trajectory of an object under gravity. For example, if the area of a square is 32 square meters, its side length \(s\) satisfies \(s^2 = 32\), which we solve as \(s = \sqrt{32} = 4\sqrt{2}\) meters. Mastering these algebraic steps builds a foundation for handling more complex physical and geometric modeling tasks.