When it comes to understanding mathematical sequences, particularly quadratic equations, things can get a bit puzzling. One common query that arises is whether a specific sequence, such as 4x^2 – 5x – 12 = 0, indeed forms a sequence. In this comprehensive guide, we will delve into this mathematical puzzle, breaking down the components, understanding its nature, and ultimately determining its validity.
Understanding Quadratic Equations
Before delving into the specific sequence provided, let’s grasp the fundamentals of quadratic equations. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (usually represented as ‘x’) is squared. The general form of a quadratic equation is:
ax^2 + bx + c = 0
Here, ‘a’, ‘b’, and ‘c’ are constants, with ‘a’ not equal to zero. The solutions to quadratic equations can be determined using various methods, such as factoring, completing the square, or using the quadratic formula.
Analyzing the Given Sequence: 4x^2 – 5x – 12 = 0
Now, let’s apply our understanding of quadratic equations to the given sequence: 4x^2 – 5x – 12 = 0. In this sequence:
- ‘a’ = 4
- ‘b’ = -5
- ‘c’ = -12
According to the quadratic equation’s standard form, the sequence represents a quadratic equation. Therefore, it’s a valid sequence to analyze further.
Determining if it’s a Sequence
To determine whether the provided sequence indeed forms a sequence, we can apply various methods, one of which is the discriminant. The discriminant of a quadratic equation is given by:
Δ = b^2 – 4ac
If the discriminant is greater than zero, the equation has two real and distinct solutions. If it equals zero, the equation has one real solution. If it’s less than zero, the equation has no real solutions.
Let’s calculate the discriminant for our sequence:
Δ = (-5)^2 – 4 * 4 * (-12)
Δ = 25 + 192
Δ = 217
Since the discriminant is greater than zero (Δ > 0), the sequence 4x^2 – 5x – 12 = 0 indeed forms a quadratic equation with two real and distinct solutions.
FAQs
Q1: What is a quadratic equation?
A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually represented as ‘x’) is squared. It has the general form: ax^2 + bx + c = 0, where ‘a’, ‘b’, and ‘c’ are constants, with ‘a’ not equal to zero.
Q2: How do you determine if a sequence forms a quadratic equation?
To determine if a sequence forms a quadratic equation, check if it follows the general form of a quadratic equation: ax^2 + bx + c = 0. If it does, where ‘a’, ‘b’, and ‘c’ are constants, then it’s a quadratic equation.
Q3: What is the discriminant of a quadratic equation?
The discriminant of a quadratic equation is a value calculated using the formula: Δ = b^2 – 4ac. It helps determine the nature of the solutions of the quadratic equation. If Δ > 0, the equation has two real and distinct solutions. If Δ = 0, the equation has one real solution. If Δ < 0, the equation has no real solutions.
Q4: What does it mean if the discriminant is greater than zero?
If the discriminant (Δ) of a quadratic equation is greater than zero, it indicates that the equation has two real and distinct solutions. This means there are two different values of ‘x’ that satisfy the equation.
Conclusion
In conclusion, after thorough analysis, it’s evident that the sequence 4x^2 – 5x – 12 = 0 is indeed a quadratic equation. By applying the quadratic formula or other suitable methods, the solutions to this equation can be accurately determined. Understanding the nature of quadratic equations and how to analyze them empowers individuals to tackle mathematical challenges with confidence and clarity.
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