91Ó°ÊÓ

When solving quadratic equations such as the one given, the zero-product property is a critical tool. This property states that if the product of two numbers is zero, then at least one of the numbers must be zero. This is why it's so effective for equations set to zero. Given any quadratic equation in the form of

• \(a \cdot b = 0\)

you can determine that either \(a = 0\) or \(b = 0\), or both. Applying this property to the equation \(x(x+2)=0\), we can split it into two separate cases:

• \(x = 0\)
• \(x+2 = 0\)

This allows us to solve for \(x\) by dealing with simpler linear equations, facilitating straightforward solutions for each possible value of \(x\).

Algebraic solutions involve using the principles of algebra to find values that satisfy an equation. In the exercise provided, we are given a product of two expressions \(x(x+2)\) set equal to zero. Solving this algebraically, we can apply the zero-product property to break it down into two separate equations:

• \(x = 0\)
• \(x + 2 = 0\)

Each simplified equation represents a potential solution. The first step in algebraically solving these is recognizing that the first equation \(x = 0\) requires no further manipulation; it is already a solution.
The second equation \(x + 2 = 0\) requires an algebraic operation: subtraction of 2 from both sides. This yields the solution \(x = -2\). Thus, using algebraic solutions, we find the complete solution set for the original equation: \(x = 0\) and \(x = -2\).

Solving quadratic equations often involves several steps like setting the equation to zero, factoring, or applying specific algebraic rules. Each method aims to simplify the equation until it is solvable via simple arithmetic operations. In this particular case, we started with the equation:

• \(x(x+2) = 0\)

To solve it,

1. First, recognize the structure of the equation where the zero-product property applies.
2. Secondly, break the equation into simpler components that involve solving basic linear equations.

By applying these steps, we find that \(x = 0\) or \(x = -2\) are the solutions as these are the values of \(x\) that make the original equation equal zero. Solving equations is about simplifying complex problems into easier, solvable steps and using properties like the zero-product property to achieve this.