(x-5).jpeg "(x-3)(x-5)")
Understanding (x-3)(x-5)
The expression (x-3)(x-5) represents the product of two binomials. Let's explore what this means and how to simplify it.
Expanding the Expression
To simplify (x-3)(x-5), we use the distributive property (also known as FOIL):
- First: Multiply the first terms of each binomial: x * x = x²
- Outer: Multiply the outer terms: x * -5 = -5x
- Inner: Multiply the inner terms: -3 * x = -3x
- Last: Multiply the last terms: -3 * -5 = 15
Now, combine the terms: x² - 5x - 3x + 15
Finally, simplify by combining like terms: x² - 8x + 15
The Resulting Quadratic Equation
The simplified expression, x² - 8x + 15, is a quadratic equation. It represents a parabola when graphed, and it can be factored back into its original form: (x-3)(x-5).
Applications of (x-3)(x-5)
This expression can be used in various mathematical contexts, such as:
- Solving quadratic equations: Setting the expression equal to zero and solving for x will provide the roots of the equation.
- Finding the zeros of a function: The zeros of a function are the values of x where the function equals zero.
- Analyzing the behavior of a parabola: The expression helps determine the vertex, axis of symmetry, and intercepts of the parabola.
Conclusion
Understanding how to expand and simplify expressions like (x-3)(x-5) is essential for working with quadratic equations and functions. It provides valuable insights into the behavior of these expressions and their applications in various mathematical contexts.