Quadratic Equation (2x-1)(x+6)=0 Standard Form And Value Of B

Introduction to Quadratic Equations

In the realm of algebra, quadratic equations stand as fundamental tools for modeling a myriad of real-world phenomena, ranging from the trajectory of projectiles to the intricate curves of suspension bridges. These equations, characterized by their second-degree polynomial nature, possess a distinctive standard form that serves as a cornerstone for analysis and problem-solving. This exploration delves into the process of transforming a given quadratic equation into its standard form and subsequently extracting the coefficient 'b', a crucial parameter that dictates the equation's behavior and properties.

Understanding the Significance of Standard Form

The standard form of a quadratic equation is expressed as ax² + bx + c = 0, where 'a', 'b', and 'c' represent constant coefficients, and 'x' denotes the variable. This particular arrangement unlocks a treasure trove of analytical possibilities. It allows us to readily identify the coefficients, which are instrumental in employing various techniques such as factoring, completing the square, and the quadratic formula to determine the equation's solutions, also known as roots or zeros. Moreover, the standard form provides valuable insights into the parabola's shape, orientation, and key features when the quadratic equation is graphically represented.

The Importance of the 'b' Coefficient

Among the coefficients, 'b' holds a special significance. It plays a pivotal role in determining the axis of symmetry of the parabola, which is a vertical line that divides the parabola into two symmetrical halves. The axis of symmetry is located at x = -b/(2a), highlighting the direct influence of 'b' on the parabola's horizontal positioning. Additionally, the 'b' coefficient contributes to the vertex's coordinates, which represents the parabola's minimum or maximum point. Understanding 'b' is crucial for sketching the parabola's graph and predicting the equation's behavior.

Transforming (2x-1)(x+6)=0 into Standard Form

Let's embark on the journey of transforming the given quadratic equation, (2x-1)(x+6)=0, into its standard form. This transformation involves a series of algebraic manipulations that meticulously expand and rearrange the equation to conform to the ax² + bx + c = 0 template.

Step 1: Expanding the Product

The initial step entails expanding the product of the two binomials, (2x-1) and (x+6). This expansion is achieved through the application of the distributive property, also known as the FOIL (First, Outer, Inner, Last) method. By meticulously multiplying each term in the first binomial with each term in the second binomial, we obtain:

(2x-1)(x+6) = 2x * x + 2x * 6 - 1 * x - 1 * 6

Step 2: Simplifying the Expression

Next, we simplify the expression by performing the multiplications and combining like terms:

2x * x + 2x * 6 - 1 * x - 1 * 6 = 2x² + 12x - x - 6

Combining the 'x' terms, we get:

2x² + 12x - x - 6 = 2x² + 11x - 6

Step 3: Expressing in Standard Form

Now, we have the expanded form of the equation: 2x² + 11x - 6 = 0. This expression perfectly aligns with the standard form ax² + bx + c = 0. We can clearly identify the coefficients:

a = 2 b = 11 c = -6

Identifying the Value of 'b'

Having successfully transformed the equation into standard form, the task of identifying the value of 'b' becomes straightforward. By comparing our expanded equation, 2x² + 11x - 6 = 0, with the standard form ax² + bx + c = 0, we can directly extract the coefficient of the 'x' term.

The Value of 'b'

Therefore, the value of 'b' in the quadratic equation (2x-1)(x+6)=0 is 11. This value, as discussed earlier, plays a significant role in determining the parabola's axis of symmetry and the vertex's coordinates.

Significance of 'b' in the Quadratic Equation

The coefficient 'b' in a quadratic equation, as mentioned earlier, is not just a numerical value; it carries significant implications for the equation's graphical representation and overall behavior. Understanding the role of 'b' is crucial for gaining a deeper comprehension of quadratic functions and their applications.

Axis of Symmetry and the 'b' Coefficient

The axis of symmetry, a vertical line that divides the parabola into two symmetrical halves, is directly influenced by the 'b' coefficient. The formula for the axis of symmetry is given by x = -b/(2a), where 'a' is the coefficient of the x² term. This formula reveals that the 'b' coefficient, in conjunction with 'a', determines the horizontal position of the axis of symmetry. A larger absolute value of 'b' shifts the axis of symmetry further away from the y-axis, while the sign of 'b' dictates the direction of the shift.

Vertex Coordinates and the 'b' Coefficient

The vertex of a parabola, representing its minimum or maximum point, also depends on the 'b' coefficient. The x-coordinate of the vertex is the same as the axis of symmetry, given by x = -b/(2a). To find the y-coordinate of the vertex, we substitute this x-value back into the original quadratic equation. Thus, the 'b' coefficient plays a crucial role in determining the vertex's position, which is a key feature of the parabola.

Impact on the Parabola's Shape and Position

In essence, the 'b' coefficient influences the parabola's horizontal position and its overall shape. By understanding the relationship between 'b' and the axis of symmetry and vertex coordinates, we can gain valuable insights into the parabola's graph and its behavior. This knowledge is essential for sketching parabolas accurately and interpreting quadratic equations in various contexts.

Conclusion

In this comprehensive exploration, we have successfully navigated the process of transforming the quadratic equation (2x-1)(x+6)=0 into its standard form, ax² + bx + c = 0. Through meticulous expansion and simplification, we arrived at the standard form: 2x² + 11x - 6 = 0. From this form, we readily identified the value of 'b' as 11. Furthermore, we delved into the significance of the 'b' coefficient, highlighting its influence on the axis of symmetry, vertex coordinates, and the overall shape and position of the parabola. This understanding empowers us to analyze quadratic equations effectively and apply them to a wide range of real-world scenarios. The ability to manipulate and interpret quadratic equations is a fundamental skill in mathematics, and this exploration has equipped you with the tools and knowledge to confidently tackle such challenges.