Solve each equation by the method of your choice. $(x-3{)}^2-25=0$

Verified step by step guidance

1. Start with the given equation: \( (x-3)^2 - 25 = 0 \).
2. Isolate the squared term by adding 25 to both sides: \( (x-3)^2 = 25 \).
3. Take the square root of both sides, remembering to consider both the positive and negative roots: \( x - 3 = \pm \sqrt{25} \).
4. Simplify the square root: \( x - 3 = \pm 5 \).
5. Solve for \(x\) by adding 3 to both sides for each case: \( x = 3 + 5 \) and \( x = 3 - 5 \).

Key Concepts

Solving Quadratic Equations

Quadratic equations are polynomial equations of degree two, typically in the form ax² + bx + c = 0. Solving them involves finding values of x that satisfy the equation. Common methods include factoring, completing the square, and using the quadratic formula.

Difference of Squares

The difference of squares is a special factoring technique where an expression of the form a² - b² can be factored into (a - b)(a + b). Recognizing this pattern simplifies solving equations like (x - 3)² - 25 = 0 by rewriting 25 as 5².

Isolating the Variable

Isolating the variable means manipulating the equation to get x alone on one side. This often involves adding, subtracting, multiplying, dividing, or taking roots. It is a fundamental step in solving equations to find the exact values of the unknown.