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Factoring in math is like breaking down a big number or expression into parts that multiply together. When you have a quadratic equation like the one we are dealing with here, factoring becomes quite handy. Factoring transforms a polynomial into a product of simpler terms or 'factors'.
When you look at polynomials, you'll often see expressions like \( ax^2 + bx + c \). You're trying to find two binomials multiplied together which give you such a result. This method helps in simplifying the solving process.
For the equation \( 2x^2 + 9x - 5 = 0 \), our goal is to find two numbers that multiply to the product of \( a \) and \( c \) and add up to \( b \). These numbers will help us break down (or factor) the equation into two binomials. This results in \((2x - 1)(x + 5) = 0\) which shows it's neatly factored.

The distributive property is a useful algebraic rule that involves distributing, or sharing, a single term across terms inside parentheses. This rule is often seen as \( a(b + c) = ab + ac \).
In our example, you have \( x(2x + 9) = 5 \). Applying the distributive property helps you simplify the equation by multiplying \( x \) with each term inside the parenthesis.

• You multiply \( x \) with \( 2x \), giving \( 2x^2 \).
• Then, multiply \( x \) with \( 9 \), resulting in \( 9x \).

After applying this property, you get \( 2x^2 + 9x = 5 \). This simplifies and exposes the structure of a quadratic equation, setting it up perfectly for the next steps.

Solving equations refers to finding the values of the variables that make the equation true. In quadratic equations, solutions are often called 'roots'.
Once you have factored the quadratic equation or manipulated it into a standard form, the next step is to determine where the equation equals zero. This requires setting each factor to zero because any number multiplied by zero results in zero.
From the factored form \((2x - 1)(x + 5) = 0\), you need to individually solve:

• \(2x - 1 = 0\)
• \(x + 5 = 0\)

Solving these will give you the solutions \(x = 0.5\) and \(x = -5\). Each of these solutions satisfies the corresponding equation, proving these values are indeed the correct ones for \(x\).

Algebraic manipulation involves using algebra rules to transform expressions or equations into a simpler form or to solve them.
It starts with rearranging equations by moving terms from one side to the other. In our scenario, the given equation \( x(2x + 9) = 5 \) was simplified using the distributive property to \( 2x^2 + 9x = 5 \).
Then, you move all terms onto one side to set it to zero, leading to \( 2x^2 + 9x - 5 = 0 \). This aligns it into a form that's ready for factoring.
These manipulations apply different algebra rules to modify the structure of an equation, with each step bringing you closer to the solution. It's about maintaining balance and carefully adjusting the equation to make solving as straightforward as possible.