(x-6).jpeg "(x-1)(x-6)")
Understanding (x-1)(x-6)
The expression (x-1)(x-6) represents a product of two binomials. To understand it better, let's break it down:
Binomials
Binomials are algebraic expressions consisting of two terms. In this case, we have:
- (x-1): This binomial has two terms, 'x' and '-1'.
- (x-6): Similarly, this binomial has two terms, 'x' and '-6'.
Expanding the Expression
To expand the product, we use the distributive property (also known as FOIL method). This involves multiplying each term in the first binomial by each term in the second binomial:
- First terms: x * x = x²
- Outer terms: x * -6 = -6x
- Inner terms: -1 * x = -x
- Last terms: -1 * -6 = 6
Now, we combine the terms:
x² - 6x - x + 6
Finally, we simplify by combining like terms:
x² - 7x + 6
What does the expression represent?
The expanded expression x² - 7x + 6 represents a quadratic equation. This equation describes a parabola, which is a U-shaped curve. The roots of this equation (where the parabola crosses the x-axis) can be found by setting the expression equal to zero and solving for x.
Applications
This type of expression finds applications in various fields, including:
- Mathematics: Solving quadratic equations, finding roots, and graphing parabolas.
- Physics: Modeling projectile motion and other physical phenomena.
- Engineering: Designing structures and analyzing their stability.
By understanding the fundamentals of expanding and simplifying such expressions, you can gain valuable insights into the underlying mathematical relationships and their practical applications.