(x+1)^2-4=0

Jasonbradley

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Jun 16, 2024

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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

1. Isolate the squared term:

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

2. Take the square root of both sides:

   • Remember to consider both positive and negative roots: x + 1 = ±2

3. 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

1. Expand the square:

   • (x+1)^2 = x^2 + 2x + 1

2. Rewrite the equation:

   • x^2 + 2x + 1 - 4 = 0
   • x^2 + 2x - 3 = 0

3. Factor the quadratic expression:

   • (x + 3)(x - 1) = 0

4. 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.