(x+3)^2=-16 Square Root Property

2 min read Jun 16, 2024

Solving Quadratic Equations using the Square Root Property: (x+3)^2 = -16

The square root property is a valuable tool for solving quadratic equations in the form of (ax + b)^2 = c. This method allows us to directly isolate the variable, simplifying the solving process.

Let's walk through how to solve the equation (x + 3)^2 = -16 using this property:

1. Isolate the Squared Term

Our equation is already in the desired form, with the squared term isolated on the left side.

2. Take the Square Root of Both Sides

The square root property states that if a^2 = b, then a = ±√b. Applying this to our equation:

√((x + 3)^2) = ±√(-16)

3. Simplify

Simplifying both sides:

x + 3 = ±4i

where i is the imaginary unit, defined as √(-1).

4. Solve for x

Subtract 3 from both sides to isolate x:

x = -3 ± 4i

Solution

The solutions to the equation (x + 3)^2 = -16 are:

x = -3 + 4i
x = -3 - 4i

Key Points to Remember

Imaginary Solutions: When the square root of a negative number arises, the solutions involve imaginary numbers.
Two Solutions: The square root property always generates two solutions, one positive and one negative.

By understanding the square root property, we can efficiently solve a range of quadratic equations, including those with imaginary solutions.