Solving Quadratic Equations by Extracting Square Roots

Quadratic equations are equations that contain a squared term. They are often written in the standard form ax² + bx + c = 0, where a, b, and c are constants. There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. One method, extracting square roots, is particularly useful for solving certain types of quadratic equations.

What is Extracting Square Roots?

Extracting square roots is a method for solving quadratic equations that involves isolating the squared term and then taking the square root of both sides of the equation. This method is most effective when the quadratic equation can be written in the form x² = k, where k is a constant.

Steps for Solving Quadratic Equations by Extracting Square Roots

1. Isolate the squared term: Rearrange the equation so that the term containing the squared variable is on one side of the equation and all other terms are on the other side. For example, if the equation is 2x² - 8 = 0, you would add 8 to both sides to get 2x² = 8. Then, divide both sides by 2 to isolate the squared term, resulting in x² = 4.
2. Take the square root of both sides: Take the square root of both sides of the equation. Remember that the square root of a number has both a positive and a negative solution. For example, the square root of 4 is both 2 and -2. In the example above, taking the square root of both sides of x² = 4 gives us x = ±2.
3. Simplify: Simplify the solutions, if possible. In the example above, the solutions are x = 2 and x = -2.

Example

Let's solve the quadratic equation x² - 9 = 0 by extracting square roots.

1. Isolate the squared term: Add 9 to both sides of the equation: x² = 9.
2. Take the square root of both sides: Take the square root of both sides of the equation: x = ±3.
3. Simplify: The solutions are x = 3 and x = -3.

When to Use Extracting Square Roots

Extracting square roots is a simple and efficient method for solving quadratic equations that can be written in the form x² = k. This method is particularly useful when the quadratic equation does not have a linear term (bx = 0) or when the coefficient of the squared term is 1 (a = 1).

Conclusion

Extracting square roots is a valuable technique for solving certain types of quadratic equations. By isolating the squared term and taking the square root of both sides, you can quickly and easily find the solutions to the equation. This method is a useful addition to your toolbox for solving quadratic equations.