(x-5)^2-36=0

Jasonbradley

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Jun 17, 2024

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Solving the Quadratic Equation: (x-5)^2 - 36 = 0

This article will guide you through the steps of solving the quadratic equation (x-5)^2 - 36 = 0. We'll explore different methods and provide a clear understanding of the solution process.

Understanding the Equation

The given equation is a quadratic equation in the standard form:

ax^2 + bx + c = 0

Where:

• a = 1 (coefficient of x^2)
• b = -10 (coefficient of x)
• c = -11 (constant term)

Solving using Square Root Property

1. Isolate the squared term: Add 36 to both sides of the equation: (x-5)^2 = 36

2. Take the square root of both sides: √(x-5)^2 = ±√36 x-5 = ±6

3. Solve for x: x = 5 ± 6

4. Find the solutions: x = 5 + 6 = 11 x = 5 - 6 = -1

Therefore, the solutions to the equation (x-5)^2 - 36 = 0 are x = 11 and x = -1.

Solving by Factoring

1. Recognize the difference of squares pattern: The equation can be rewritten as: (x-5)^2 - 6^2 = 0 This resembles the difference of squares pattern: a^2 - b^2 = (a+b)(a-b)

2. Factor the equation: (x-5+6)(x-5-6) = 0 (x+1)(x-11) = 0

3. Set each factor to zero and solve: x+1 = 0 => x = -1 x-11 = 0 => x = 11

This method also confirms that the solutions are x = 11 and x = -1.

Conclusion

We have successfully solved the quadratic equation (x-5)^2 - 36 = 0 using two different methods: the square root property and factoring. Both methods demonstrate that the equation has two distinct solutions: x = 11 and x = -1.