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Solving Quadratic Equations by Extracting Square Roots

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Solving Quadratic Equations by Extracting Square Roots

Quadratic equations are equations of the form ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations are prevalent in various fields, including physics, engineering, and economics. One common method for solving quadratic equations is by extracting square roots.

What is Extracting Square Roots?

Extracting square roots is a technique that allows us to solve quadratic equations in the form of ax2 + c = 0. The method involves isolating the squared term (x2) and then taking the square root of both sides of the equation.

Steps to Solve Quadratic Equations by Extracting Square Roots

  1. Isolate the squared term: Move any constant terms to the right side of the equation.
  2. Divide both sides by the coefficient of the squared term: This ensures the coefficient of x2 is 1.
  3. Take the square root of both sides: Remember to consider both positive and negative square roots.
  4. Solve for x: Simplify the equation to find the values of x.

Example

Let’s solve the quadratic equation x2 – 9 = 0 by extracting square roots.

  1. Isolate the squared term: x2 = 9
  2. Divide both sides by the coefficient of the squared term: x2/1 = 9/1
  3. Take the square root of both sides: √x2 = ±√9
  4. Solve for x: x = ±3

Therefore, the solutions to the equation x2 – 9 = 0 are x = 3 and x = -3.

Important Notes

  • Not all quadratic equations can be solved by extracting square roots. This method works only for equations that can be rewritten in the form ax2 + c = 0.
  • Always remember to consider both positive and negative square roots when solving for x.

Practice Problems

Try solving these quadratic equations by extracting square roots:

  1. 4x2 – 16 = 0
  2. 2x2 + 18 = 0
  3. x2 – 25 = 0

Conclusion

Extracting square roots is a straightforward method for solving quadratic equations of a specific form. By following the steps outlined above, you can efficiently find the solutions to these equations. Remember to practice and apply this technique to various problems to solidify your understanding.

Algebra