91Ó°ÊÓ

Factoring is an essential skill in solving equations. It involves breaking down a complex expression into simpler terms or factors. This is important in equations where different terms have a common factor. For example, in the given equation (2-3 x) 5^x + 4 \cdot 5^x = 0, both terms have the common factor of \(5^x\). Factoring out \(5^x\) simplifies the equation as follows: \[5^x((2-3x) + 4) = 0\] Factoring makes it easier to identify solutions since each factor can be set to zero. Always remember to look for a common factor when solving polynomial equations or exponential functions. There are two types of factors here: \(5^x\) and \((2-3x) + 4\). Factoring effectively reduces the equation into simpler parts, which can be individually analyzed.

Exponential functions, such as \(5^x\), are functions where a constant base is raised to a variable exponent. These functions have unique properties that make them fundamental in algebra. The most important property is that an exponential function like \(5^x\) is never zero for any real number \(x\). In the given equation:\[5^x((2 - 3x) + 4) = 0\]One of the factors is an exponential function, \(5^x\). Due to its property, we know \(5^x eq 0\). This means any solutions to the equation must come from the other factor \((6 - 3x)\). Exponential functions grow or decay exponentially fast and knowing their properties can greatly aid in solving complex equations. When dealing with exponential functions, always recall their key characteristics, such as their non-zero nature for any real number.

Algebraic manipulation involves using algebraic techniques to simplify and solve equations. This includes combining like terms, distributing, factoring, and isolating variables. Let’s break down the manipulations step by step.First, we notice both terms have a common factor: \(5^x((2-3x) + 4) = 0\)We then combine like terms inside the parentheses:\((2-3x) + 4 = 6 - 3x\)The simplified equation becomes:\[ 5^x(6 - 3x) = 0 \]To find the values of \(x\), we set each factor to zero. Since \(5^x eq 0\), we solve \(6 - 3x = 0\)\(6 - 3x = 0\)Solving for \(x\), we find:\(-3x = -6\)\(x = 2\)These steps demonstrate the power of algebraic manipulation, especially when dealing with complex equations that can be simplified through factoring and combining like terms. Master these techniques to simplify and solve various algebraic equations more efficiently.