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Solving the Quadratic Equation: (x + 5)² - 4 = 0
This article will guide you through solving the quadratic equation (x + 5)² - 4 = 0. We'll use a step-by-step approach to find the values of x that satisfy the equation.
Step 1: Simplify the Equation
Begin by simplifying the equation:
- Expand the square: (x + 5)² = x² + 10x + 25
- Substitute the expansion: x² + 10x + 25 - 4 = 0
- Combine the constants: x² + 10x + 21 = 0
Now we have a standard quadratic equation in the form ax² + bx + c = 0, where a = 1, b = 10, and c = 21.
Step 2: Factor the Equation
To factor the equation, we need to find two numbers that add up to 10 (the coefficient of x) and multiply to 21 (the constant term). These numbers are 7 and 3:
x² + 10x + 21 = (x + 7)(x + 3) = 0
Step 3: Solve for x
For the product of two factors to be zero, at least one of them must be zero. Therefore:
- x + 7 = 0 => x = -7
- x + 3 = 0 => x = -3
Conclusion
The solutions to the quadratic equation (x + 5)² - 4 = 0 are x = -7 and x = -3.