Solving the Quadratic Equation: (x+1)^2  4 = 0
This article will explore how to solve the quadratic equation (x+1)^2  4 = 0. We'll break down the steps and use different methods to arrive at the solutions.
Understanding the Equation
The given equation is a quadratic equation in the standard form: ax^2 + bx + c = 0. In this case, we have:
 a = 1 (the coefficient of the x^2 term)
 b = 2 (the coefficient of the x term)
 c = 3 (the constant term)
Method 1: Using the Square Root Property

Isolate the squared term:
 Add 4 to both sides of the equation: (x+1)^2 = 4

Take the square root of both sides:
 Remember to consider both positive and negative roots: x + 1 = ±2

Solve for x:
 For x + 1 = 2, we get x = 1.
 For x + 1 = 2, we get x = 3.
Therefore, the solutions to the equation (x+1)^2  4 = 0 are x = 1 and x = 3.
Method 2: Expanding and Factoring

Expand the square:
 (x+1)^2 = x^2 + 2x + 1

Rewrite the equation:
 x^2 + 2x + 1  4 = 0
 x^2 + 2x  3 = 0

Factor the quadratic expression:
 (x + 3)(x  1) = 0

Set each factor to zero and solve:
 x + 3 = 0 => x = 3
 x  1 = 0 => x = 1
Again, we find the solutions x = 1 and x = 3.
Conclusion
We have successfully solved the quadratic equation (x+1)^2  4 = 0 using two different methods: the square root property and expansion/factoring. Both methods lead to the same solutions: x = 1 and x = 3. Remember to always consider both positive and negative roots when taking the square root of an equation.