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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 have two solutions, which can be found using different methods, including factoring, completing the square, and the quadratic formula. One of the simplest methods to solve certain quadratic equations is by extracting square roots.

Understanding the Method

The method of extracting square roots is based on the fundamental principle that if two quantities are equal, their square roots are also equal. This means that if a2 = b2, then a = ±b.

To solve a quadratic equation by extracting square roots, we follow these steps:

  1. Isolate the squared term: Manipulate the equation to get the term with the squared variable (x2) alone on one side of the equation.
  2. Take the square root of both sides: Take the square root of both sides of the equation, remembering to include both positive and negative roots.
  3. Solve for the variable: Simplify the equation to solve for the variable.

Examples

Example 1:

Solve the equation x2 - 9 = 0

  1. Isolate the squared term: Add 9 to both sides: x2 = 9
  2. Take the square root of both sides: √x2 = ±√9
  3. Solve for the variable: x = ±3

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

Example 2:

Solve the equation 2x2 + 8 = 0

  1. Isolate the squared term: Subtract 8 from both sides: 2x2 = -8. Divide both sides by 2: x2 = -4
  2. Take the square root of both sides: √x2 = ±√-4
  3. Solve for the variable: x = ±2i (where 'i' is the imaginary unit, √-1)

In this case, the solutions are complex numbers because the square root of a negative number is imaginary.

Limitations

The method of extracting square roots is not applicable to all quadratic equations. It is most effective when the equation has no linear term (bx = 0). If the equation has a linear term, you can use other methods like factoring or the quadratic formula.

Conclusion

Extracting square roots is a simple and efficient method for solving quadratic equations in a specific form. Understanding this method can help you gain a deeper understanding of quadratic equations and their solutions. Remember to always consider both positive and negative square roots when applying this technique.

quadratic equations