(x-3).jpeg "(x-7)(x-3)")
Expanding and Understanding (x-7)(x-3)
This expression represents the product of two binomials: (x-7) and (x-3). To understand it better, we can expand it using the FOIL method.
Expanding using FOIL
FOIL stands for First, Outer, Inner, Last. It's a mnemonic to help remember the order of multiplication for expanding two binomials:
- First: Multiply the first terms of each binomial: x * x = x²
- Outer: Multiply the outer terms of the binomials: x * -3 = -3x
- Inner: Multiply the inner terms of the binomials: -7 * x = -7x
- Last: Multiply the last terms of each binomial: -7 * -3 = 21
Now, combining the terms we get: x² - 3x - 7x + 21
Finally, combining like terms: x² - 10x + 21
Therefore, the expanded form of (x-7)(x-3) is x² - 10x + 21.
Understanding the Result
This expanded form represents a quadratic expression. Its graph is a parabola, and its roots are the values of x where the expression equals zero. These roots can be found by factoring the expression:
(x-7)(x-3) = 0
This equation is true when either (x-7) = 0 or (x-3) = 0. Solving for x, we get x = 7 and x = 3.
These are the roots of the quadratic expression and represent the x-intercepts of its graph.
Applications
This simple expression has various applications in mathematics and other fields:
- Solving Equations: It can be used to solve quadratic equations by factoring.
- Graphing Functions: Understanding the roots and the shape of the parabola helps in graphing the function represented by the expression.
- Modeling Real-World Phenomena: Quadratic expressions can model various real-world scenarios, such as projectile motion and optimization problems.
Understanding the expansion and the roots of this simple expression helps build a foundation for further exploration and application of quadratic equations in various contexts.