(x - 6)^2 - 5 = 0

Jasonbradley

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Jun 16, 2024

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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.

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

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

  3. 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 x-axis.