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(x-4)2-25=0 Extracting Square Root

3 min read Jun 17, 2024
(x-4)2-25=0 Extracting Square Root

Solving Quadratic Equations by Extracting Square Roots

This article will guide you through solving the quadratic equation (x - 4)² - 25 = 0 by using the method of extracting square roots. This method is particularly useful for equations where the quadratic term is a perfect square and the linear term is absent.

Steps to solve the Equation:

  1. Isolate the squared term: Begin by adding 25 to both sides of the equation to isolate the term with the squared expression. This gives us: (x - 4)² = 25

  2. Take the square root of both sides: The next step is to take the square root of both sides of the equation. Remember that taking the square root of a number can result in both a positive and a negative value. Therefore: √(x - 4)² = ±√25

  3. Simplify: Simplify the square roots on both sides of the equation. This gives us: x - 4 = ±5

  4. Solve for x: To find the possible values of x, we need to isolate x. Add 4 to both sides of the equation: x = 4 ± 5

  5. Calculate the solutions: Finally, we have two possible solutions for x: x = 4 + 5 = 9 x = 4 - 5 = -1

Therefore, the solutions to the quadratic equation (x - 4)² - 25 = 0 are x = 9 and x = -1.

Advantages of the Extracting Square Roots Method:

  • Simplicity: It provides a straightforward and efficient method for solving specific quadratic equations.
  • Direct Solution: It directly leads to the solutions without the need for factorization or the quadratic formula.

Limitations of the Extracting Square Roots Method:

  • Specific Form: This method only works for quadratic equations where the quadratic term is a perfect square and the linear term is absent.

By following these steps, you can successfully solve quadratic equations like (x - 4)² - 25 = 0 using the extracting square roots method. Remember to be mindful of the positive and negative values when taking the square root.

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