Solving Quadratic Equations by Extracting Square Roots

Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable is 2. They often appear in various fields like physics, engineering, and finance. One method for solving quadratic equations is by extracting square roots. This method is particularly useful when the equation is in a specific form, making it simpler than factoring or using the quadratic formula.

Understanding the Method

The method of extracting square roots relies on the principle of isolating the squared term and then taking the square root of both sides of the equation. This process leads to two possible solutions, as the square root of a number can be both positive and negative.

Steps to Solve by Extracting Square Roots

Here’s a step-by-step guide to solving quadratic equations using the extraction of square roots method:

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.
2. Divide by the coefficient of the squared term: Ensure the coefficient of the squared term is 1. Divide both sides of the equation by the coefficient of the squared term.
3. Take the square root of both sides: Apply the square root operation to both sides of the equation. Remember that taking the square root results in both positive and negative solutions.
4. Solve for the variable: Simplify the equation and solve for the variable. You will obtain two solutions, one positive and one negative.

Example

Let’s solve the quadratic equation x² – 9 = 0 using the extraction of square roots method:

1. Isolate the squared term: We already have the squared term isolated: x² = 9.
2. Divide by the coefficient of the squared term: The coefficient of x² is already 1, so no division is necessary.
3. Take the square root of both sides: √x² = ±√9.
4. Solve for the variable: x = ±3.

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

Important Notes

• The extraction of square roots method is only applicable to quadratic equations where the linear term (the term with the variable to the power of 1) is absent.
• If the constant term on the right side of the equation is negative, the equation will have no real solutions, as the square root of a negative number is imaginary.

Conclusion

Extracting square roots provides a straightforward method for solving specific quadratic equations. By understanding the steps involved and the limitations of this method, you can effectively apply it to solve relevant problems in various fields.