(x%2B2).jpeg "(x-4)(x+2)")
Factoring and Solving the Expression (x-4)(x+2)
The expression (x-4)(x+2) represents a product of two binomials. It's important to understand how to factor and solve such expressions in algebra.
Factoring the Expression
Factoring means breaking down an expression into its simplest components, often by finding factors that multiply to give the original expression. In this case, the expression is already factored. The factors are:
- (x-4): This represents the first binomial factor.
- (x+2): This represents the second binomial factor.
Expanding the Expression
If we want to find the expanded form of the expression, we need to multiply the two binomials using the distributive property (also known as FOIL):
First: x * x = x² Outer: x * 2 = 2x Inner: -4 * x = -4x Last: -4 * 2 = -8
Combining like terms, we get: x² + 2x - 4x - 8 = x² - 2x - 8
Therefore, the expanded form of the expression (x-4)(x+2) is x² - 2x - 8.
Solving for x
To solve for x, we need to find the values of x that make the expression equal to zero. This means finding the roots or zeros of the equation:
(x-4)(x+2) = 0
Using the zero product property, we know that for the product of two factors to be zero, at least one of them must be zero. Therefore, we have two possible solutions:
- x - 4 = 0 => x = 4
- x + 2 = 0 => x = -2
Therefore, the solutions to the equation (x-4)(x+2) = 0 are x = 4 and x = -2.
Conclusion
The expression (x-4)(x+2) is a factored form of a quadratic expression. We can expand it to find the equivalent expression x² - 2x - 8. Solving for x involves finding the roots of the equation, which are x = 4 and x = -2. Understanding how to factor, expand, and solve expressions like this is crucial in algebra and other areas of mathematics.