(x%2B3).jpeg "(x-7)(x+3)")
Factoring and Solving the Expression: (x-7)(x+3)
This expression represents a factored form of a quadratic equation. Let's break down its components and understand what it means.
Understanding the Factored Form
The expression (x-7)(x+3) is in factored form. This means that it represents the product of two binomials: (x-7) and (x+3).
Key Points about Factoring:
- Binomials: These are algebraic expressions with two terms.
- Factored Form: This represents a product of expressions, making it easier to solve for the values of the variable (x in this case).
Expanding the Expression
To understand the original quadratic equation, we need to expand the expression:
(x-7)(x+3) = x(x+3) - 7(x+3) = x² + 3x - 7x - 21 = x² - 4x - 21
Therefore, the expanded form of the expression is x² - 4x - 21.
Finding the Solutions (Roots)
The factored form allows us to find the solutions (roots) of the quadratic equation easily. To find the roots, we set each binomial factor equal to zero and solve for x:
- x - 7 = 0
- x = 7
- x + 3 = 0
- x = -3
Therefore, the solutions (roots) of the equation (x-7)(x+3) = 0 are x = 7 and x = -3.
Conclusion
The expression (x-7)(x+3) is a factored quadratic equation. Expanding it gives us the standard form x² - 4x - 21. By setting each factor equal to zero, we find the solutions to the equation, which are x = 7 and x = -3. Understanding the concept of factoring and its relationship to solving quadratic equations is crucial in algebra.