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(x−6)^2−5=0

3 min read Jun 17, 2024
(x−6)^2−5=0

Solving the Quadratic Equation: (x-6)^2 - 5 = 0

This article will walk you through the steps to solve the quadratic equation (x-6)^2 - 5 = 0. We'll explore two methods: taking the square root and using the quadratic formula.

Method 1: Taking the Square Root

  1. Isolate the squared term:

    • Add 5 to both sides of the equation: (x-6)^2 = 5

  2. Take the square root of both sides:

    • Remember to include both positive and negative square roots: x - 6 = ±√5

  3. Solve for x:

    • Add 6 to both sides: x = 6 ±√5

Therefore, the solutions to the equation are x = 6 + √5 and x = 6 - √5.

Method 2: Using the Quadratic Formula

  1. Expand the squared term:

    • (x-6)^2 = x^2 - 12x + 36
    • The equation becomes: x^2 - 12x + 36 - 5 = 0
    • Simplify: x^2 - 12x + 31 = 0

  2. Identify coefficients:

    • In the standard quadratic equation ax^2 + bx + c = 0, we have:
      • a = 1
      • b = -12
      • c = 31

  3. Apply the quadratic formula:

    • x = (-b ± √(b^2 - 4ac)) / 2a
    • Substitute the values: x = (12 ± √((-12)^2 - 4 * 1 * 31)) / 2 * 1
    • Simplify: x = (12 ± √(144 - 124)) / 2
    • Further simplification: x = (12 ± √20) / 2
    • x = (12 ± 2√5) / 2

  4. Simplify the solution:

    • x = 6 ± √5

We arrive at the same solutions as before: x = 6 + √5 and x = 6 - √5.

Conclusion

Both methods effectively solve the quadratic equation (x-6)^2 - 5 = 0, resulting in the same solutions: x = 6 + √5 and x = 6 - √5. Choosing the appropriate method depends on your preference and the complexity of the equation.

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