Solving the Quadratic Equation: (x  6)^2  5 = 0
This article will walk you through the steps of solving the quadratic equation (x  6)^2  5 = 0.
Understanding the Equation
The equation (x  6)^2  5 = 0 is a quadratic equation in standard form. This means it can be written as ax^2 + bx + c = 0, where a, b, and c are constants. In our case, we can expand the equation to get:
x^2  12x + 36  5 = 0 x^2  12x + 31 = 0
Solving the Equation
There are several methods to solve quadratic equations, including:

Factoring: This method involves finding two numbers that multiply to give the constant term (31) and add up to the coefficient of the linear term (12). Unfortunately, this equation doesn't factor easily.

Completing the Square: This method involves manipulating the equation to create a perfect square trinomial.

Move the constant term to the right side: (x  6)^2 = 5

Take the square root of both sides: x  6 = ±√5

Isolate x: x = 6 ± √5


Quadratic Formula: This method provides a direct solution for any quadratic equation. The quadratic formula is:
x = (b ± √(b^2  4ac)) / 2a
In our equation, a = 1, b = 12, and c = 31. Substituting these values into the formula:
x = (12 ± √((12)^2  4 * 1 * 31)) / (2 * 1)
x = (12 ± √(144  124)) / 2
x = (12 ± √20) / 2
x = (12 ± 2√5) / 2
x = 6 ± √5
Solutions
Therefore, the solutions to the quadratic equation (x  6)^2  5 = 0 are:
 x = 6 + √5
 x = 6  √5
These represent the two points where the graph of the equation intersects the xaxis.