%5E2-expand.jpeg "(x-4)^2 Expand")
Expanding (x-4)^2
The expression (x-4)^2 represents the square of the binomial (x-4). To expand this expression, we can use the following methods:
1. Using the FOIL Method:
FOIL stands for First, Outer, Inner, Last. This method is a systematic way to multiply two binomials.
- First: Multiply the first terms of each binomial: x * x = x^2
- Outer: Multiply the outer terms of the binomials: x * -4 = -4x
- Inner: Multiply the inner terms of the binomials: -4 * x = -4x
- Last: Multiply the last terms of each binomial: -4 * -4 = 16
Now, combine all the terms: x^2 - 4x - 4x + 16
Finally, simplify by combining like terms: x^2 - 8x + 16
2. Using the Square of a Binomial Formula:
The formula for squaring a binomial is: (a - b)^2 = a^2 - 2ab + b^2
In our case, a = x and b = 4. Substituting into the formula:
(x - 4)^2 = x^2 - 2(x)(4) + 4^2
Simplifying the expression: x^2 - 8x + 16
Understanding the Expansion:
Expanding (x-4)^2 gives us a quadratic expression: x^2 - 8x + 16. This represents a parabola when graphed. The expansion reveals the following:
- x^2: The leading term determines the shape and direction of the parabola.
- -8x: This term contributes to the slope of the parabola.
- 16: This term represents the y-intercept, where the parabola crosses the y-axis.
Applications of Expanding (x-4)^2:
Expanding (x-4)^2 is useful in various mathematical applications, such as:
- Solving quadratic equations: The expanded form can be used to solve for the roots of the equation.
- Graphing parabolas: The expanded form provides information about the shape, direction, and position of the parabola.
- Finding maximum or minimum values: The expanded form can be used to find the vertex of the parabola, which represents the maximum or minimum value of the function.
By understanding the different methods of expanding (x-4)^2, you gain a deeper understanding of its mathematical properties and its applications in various contexts.