Evaluate 3a2 4b2 for a 2 and b 3 – Embark on a mathematical journey as we delve into the intricacies of “evaluate 3a2 4b2 for a = 2 and b = 3.” This exploration promises to unveil the secrets behind this enigmatic expression, revealing its numerical value, algebraic simplification, graphical representation, and real-world applications.
Join us as we unravel the mysteries of 3a2 4b2, step by step, uncovering the beauty and practicality of mathematical expressions.
Mathematical Expression
The expression “evaluate 3a 24b 2for a = 2 and b = 3″ means to find the numerical value of the expression when a is replaced by 2 and b is replaced by 3.
Step-by-Step Process
- Substitute a = 2 and b = 3 into the expression: 3(2)24(3) 2
- Simplify the squares: 3(4) 4(9)
- Perform the multiplication: 12 36
- Multiply the two numbers: 12 x 36 = 432
Mathematical Operations
The mathematical operations involved in the evaluation are:
- Substitution
- Squaring
- Multiplication
Numerical Evaluation
Let’s evaluate the numerical value of the expression “3a2 4b2” when a = 2 and b = 3. This involves substituting the values of a and b into the expression and performing the necessary calculations.
Numerical Evaluation
Substituting a = 2 and b = 3 into the expression, we get:
3a2 4b2 = 3(2)2 4(3)2
Simplifying further:
3a2 4b2 = 3(4) 4(9)
3a2 4b2 = 12 36
Therefore, the numerical value of the expression “3a2 4b2” when a = 2 and b = 3 is 48.
Algebraic Simplification
Algebraic simplification is the process of transforming an algebraic expression into an equivalent expression that is simpler or easier to understand. In this section, we will simplify the expression “3a2 4b2” algebraically.
Identifying Common Factors
The first step in algebraic simplification is to identify any common factors in the expression. Common factors are terms that are multiplied together in each term of the expression. In this case, the common factor is “2”.
Factoring Out the Common Factor
Once we have identified the common factor, we can factor it out of the expression. To do this, we divide each term of the expression by the common factor.
“`
a2 4b2 = 2(3a2/2) 2(4b2/2)
“`
Simplifying the Expression, Evaluate 3a2 4b2 for a 2 and b 3
We can now simplify the expression by dividing each term by the corresponding denominator.
“`
a2 4b2 = 2(3a2/2) 2(4b2/2)
= 2(3a) 2(2b)= 6a2b“`
Therefore, the simplified expression is 6a2b.
Graphing the Expression
The expression 3a 24b 2can be graphed in a 3D coordinate system, where the x-axis represents the value of a, the y-axis represents the value of b, and the z-axis represents the value of the expression.
The graph of the expression is a paraboloid that opens upwards. The vertex of the paraboloid is at the origin (0, 0, 0), and the axis of symmetry is the z-axis.
Shape and Characteristics of the Graph
- The graph is symmetric with respect to the z-axis.
- The graph is unbounded in all directions.
- The graph has a minimum value of 0 at the vertex.
- The graph increases without bound as a and b increase.
Applications in Real-World Scenarios: Evaluate 3a2 4b2 For A 2 And B 3
The expression “3a2 4b2” finds applications in various real-world scenarios. It is commonly used in physics, engineering, and other scientific disciplines to model physical phenomena and solve practical problems.
Engineering and Construction
In engineering and construction, the expression “3a2 4b2” is used to calculate the area of an ellipse. An ellipse is a two-dimensional shape that resembles a stretched circle. It is often encountered in the design of bridges, arches, and other architectural structures.
The area of an ellipse is given by the formula A = πab, where ‘a’ and ‘b’ represent the lengths of the ellipse’s semi-major and semi-minor axes, respectively. By substituting ‘a’ with ‘2a’ and ‘b’ with ‘3b’, we obtain the expression “3a2 4b2”.
This expression allows engineers to determine the area of an ellipse based on the lengths of its axes.
Question & Answer Hub
What is the numerical value of 3a2 4b2 for a = 2 and b = 3?
The numerical value is 72.
Can 3a2 4b2 be simplified algebraically?
Yes, it can be simplified to 12(a2 + b2).
