%5E2.jpeg "(x-8)^2")
Understanding (x-8)^2
The expression (x-8)^2 represents the square of the binomial (x-8). Understanding how to expand and simplify this expression is fundamental in algebra and has various applications in different fields.
Expanding the Expression
To expand (x-8)^2, we can use the FOIL method or the square of a binomial formula:
FOIL Method:
- First: x * x = x^2
- Outer: x * -8 = -8x
- Inner: -8 * x = -8x
- Last: -8 * -8 = 64
Therefore, (x-8)^2 = x^2 - 8x - 8x + 64
Square of a Binomial Formula:
The formula states: (a-b)^2 = a^2 - 2ab + b^2
Applying this to our expression:
(x-8)^2 = x^2 - 2(x)(8) + 8^2 = x^2 - 16x + 64
Simplifying the Expression
In both cases, we arrive at the simplified form: x^2 - 16x + 64
Applications
This expression can be used in various contexts, including:
- Solving quadratic equations: Setting (x-8)^2 equal to a constant value leads to a quadratic equation that can be solved for x.
- Graphing parabolas: The expression represents a parabola opening upwards with its vertex at (8, 0).
- Calculus: The derivative of (x-8)^2 is 2x-16, which can be used to find the slope of the tangent line at any point on the curve.
Conclusion
Understanding the expansion and simplification of (x-8)^2 is crucial for mastering algebraic concepts and its applications. By applying the FOIL method or the square of a binomial formula, we can easily simplify the expression and use it in various mathematical contexts.