Quadratic Equations

The Square Root Property

Quadratic equations are equations of the form . While factoring is a common method for solving quadratics, it is not always possible. The Square Root Property provides an alternative approach, especially when the equation can be written in the form .

• Square Root Property: If , then .

• This method is useful when the quadratic is not easily factorable.

• Other methods for solving quadratics include completing the square and the quadratic formula.

Methods for Solving Quadratic Equations

Examples

• Example 1: Solve using the square root property.

  • Isolate :

  • Apply the square root property:

  • Express as complex numbers:

• Example 2: Solve using the square root property.

  • Isolate :

  • Apply the square root property:

Imaginary Roots

When the solution to yields a negative value for , the roots are imaginary or complex. These are written in terms of , where .

• Example: Solve using the square root property.

  • Isolate :

  • Apply the square root property:

Steps for Using the Square Root Property

1. Isolate the squared term ().

2. Take the square root of both sides.

3. Include both the positive and negative roots ().

4. Simplify and check the solution.

Note: If and have the same sign in the standard form , you will always end up with a complex answer.