What is the expression in factored form 4×2 11x 6 – In the realm of algebra, the expression 4x^2 + 11x + 6 takes center stage, inviting us on an intellectual journey to unravel its intricacies. This expression, when factored into its simplest form, reveals a deeper understanding of its mathematical structure and opens up a world of practical applications.
Embarking on this exploration, we will delve into the concept of factoring quadratic expressions, uncover the various methods employed, and ultimately grasp the significance of the factored form in solving real-world problems.
Expression Analysis
The expression “4x^2 + 11x + 6” is a quadratic expression, which is a polynomial of degree 2. It represents a parabola that opens upward or downward, depending on the sign of the leading coefficient (4 in this case).
To factor this expression, we can use the following steps:
- Find two numbers that multiply to give the constant term (6) and add to give the coefficient of the middle term (11). In this case, the numbers are 2 and 3.
- Rewrite the middle term using these two numbers: 11x = 2x + 3x.
- Factor by grouping: (4x^2 + 2x) + (3x + 6) = 2x(2x + 1) + 3(2x + 1).
- Factor out the common factor (2x + 1): (2x + 1)(2x + 3).
Factoring Methods
There are several methods for factoring quadratic expressions:
- Factoring by Grouping:This method is used when the coefficient of the middle term is not easily factorable.
- Using the Quadratic Formula:This method is a general formula that can be used to factor any quadratic expression.
In this case, we can use the factoring by grouping method, as shown in the previous section.
Factored Form
The factored form of a quadratic expression is an expression that has been factored into its simplest form, which is usually the product of two linear factors.
The factored form of “4x^2 + 11x + 6” is (2x + 1)(2x + 3).
Applications of Factoring: What Is The Expression In Factored Form 4×2 11x 6
Factoring quadratic expressions has many practical applications, including:
- Solving Equations:Factoring can be used to solve quadratic equations by setting each factor equal to zero and solving for the variable.
- Graphing Parabolas:The factored form of a quadratic expression can be used to graph the parabola it represents by identifying the vertex and axis of symmetry.
- Solving Real-World Problems:Quadratic equations can be used to model a wide variety of real-world problems, such as projectile motion, area, and volume.
For example, we can use factoring to solve the equation 4x^2 + 11x + 6 = 0 by setting each factor equal to zero:
- 2x + 1 = 0, which gives x = -1/2
- 2x + 3 = 0, which gives x = -3/2
Therefore, the solutions to the equation are x = -1/2 and x = -3/2.
FAQ Resource
What is the factored form of 4x^2 + 11x + 6?
(2x + 3)(2x + 2)
How can I factor quadratic expressions?
There are several methods, including factoring by grouping and using the quadratic formula.
What are the applications of factoring quadratic expressions?
Solving equations, graphing parabolas, and optimizing functions.
