Finding The Roots Of 6x^2 - X - 2 = 0

When we talk about the roots of an equation, especially a quadratic one like $6x^2 - x - 2 = 0$, we're essentially looking for the values of $x$ that make the equation true. Think of it like finding the specific points where a parabola (the graph of a quadratic equation) crosses the x-axis. These roots are also often referred to as the solutions or zeros of the equation. For the equation $6x^2 - x - 2 = 0$, there are a couple of common and effective methods to uncover these crucial values. One of the most straightforward approaches is factoring, which involves breaking down the quadratic expression into a product of two linear expressions. Another powerful technique is the quadratic formula, a universal tool that can solve any quadratic equation, no matter how complex. Let's dive into these methods to find the roots of $6x^2 - x - 2 = 0$ and understand what they represent.

Method 1: Factoring the Quadratic Equation

Factoring is often the most elegant way to find the roots of a quadratic equation if it's possible. For our equation, $6x^2 - x - 2 = 0$, we need to find two binomials that, when multiplied together, yield this exact expression. This process can sometimes feel a bit like detective work. We're looking for two numbers that multiply to give us the product of the leading coefficient (6) and the constant term (-2), which is $6 \times -2 = -12$. Simultaneously, these same two numbers must add up to the coefficient of the middle term, which is -1. Let's brainstorm pairs of numbers that multiply to -12: (1, -12), (-1, 12), (2, -6), (-2, 6), (3, -4), and (-3, 4). Now, let's check which of these pairs sums to -1. We can see that 3 and -4 fit the bill, as $3 \times -4 = -12$ and $3 + (-4) = -1$. With these magic numbers identified, we can rewrite the middle term, $-x$, as $3x - 4x$. So, our equation transforms into $6x^2 + 3x - 4x - 2 = 0$. The next step is to group the terms: $(6x^2 + 3x) + (-4x - 2) = 0$. Now, we factor out the greatest common factor (GCF) from each group. In the first group, the GCF is $3x$, leaving us with $3x(2x + 1)$. In the second group, the GCF is -2, which, when factored out, gives us $-2(2x + 1)$. Notice that we now have a common binomial factor of $(2x + 1)$ in both terms: $3x(2x + 1) - 2(2x + 1) = 0$. We can now factor out this common binomial: $(2x + 1)(3x - 2) = 0$. For the product of these two factors to be zero, at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for $x$: $2x + 1 = 0$ or $3x - 2 = 0$. Solving the first equation, $2x = -1$, which gives us $x = -1/2$. Solving the second equation, $3x = 2$, which gives us $x = 2/3$. So, the roots of the equation $6x^2 - x - 2 = 0$ are $x = -1/2$ and $x = 2/3$. These are the specific values of $x$ that will make the original equation hold true. Understanding factoring is a key skill in algebra, and it's particularly useful for simplifying and solving polynomial equations. The ability to recognize and apply factoring techniques can significantly speed up problem-solving and deepen your comprehension of algebraic structures.

Method 2: Using the Quadratic Formula

The quadratic formula is a guaranteed method to find the roots of any quadratic equation in the standard form $ax^2 + bx + c = 0$. It's a lifesaver when factoring seems impossible or too time-consuming. The formula itself is derived from the process of completing the square on the general quadratic equation and is given by: $x = \frac-b \pm \sqrt{b^2 - 4ac}}{2a}$ For our specific equation, $6x^2 - x - 2 = 0$, we first need to identify the coefficients $a$, $b$, and $c$. In this case, $a = 6$ (the coefficient of $x^2$), $b = -1$ (the coefficient of $x$), and $c = -2$ (the constant term). Now, we substitute these values into the quadratic formula $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(6)(-2)}2(6)}$ Let's simplify this expression step by step. First, the term $-(-1)$ becomes $1$. The term $(-1)^2$ is $1$. The term $4(6)(-2)$ is $4 \times -12$, which equals $-48$. So, the expression under the square root (the discriminant) becomes $1 - (-48) = 1 + 48 = 49$. The denominator $2(6)$ is $12$. Thus, the formula simplifies to $x = \frac{1 \pm \sqrt{49}12}$ The square root of 49 is 7. So, we have $x = \frac{1 \pm 7{12}$ This $\pm$ symbol indicates that there are two possible solutions. We calculate each one separately:

1. Using the plus sign: $x_1 = \frac{1 + 7}{12} = \frac{8}{12}$. This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 4. So, $x_1 = \frac{8 \div 4}{12 \div 4} = \frac{2}{3}$.
2. Using the minus sign: $x_2 = \frac{1 - 7}{12} = \frac{-6}{12}$. This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 6. So, $x_2 = \frac{-6 \div 6}{12 \div 6} = -\frac{1}{2}$.

As you can see, the quadratic formula gives us the exact same roots as the factoring method: $x = 2/3$ and $x = -1/2$. The quadratic formula is incredibly powerful because it works for all quadratic equations, even those where the roots are irrational or complex numbers, which would be impossible to find by factoring with integers. When dealing with the quadratic formula, pay close attention to the signs of $a$, $b$, and $c$, as a misplaced sign can lead to an incorrect result. The discriminant ($b^2 - 4ac$) is particularly important as it tells us the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is exactly one real root (a repeated root); and if it's negative, there are two complex conjugate roots.

Understanding the Roots and Their Significance

Now that we've found the roots of the equation $6x^2 - x - 2 = 0$ using two different methods, it's important to understand what these numbers, $x = 2/3$ and $x = -1/2$, truly signify. In the realm of mathematics, these roots represent the x-intercepts of the parabola defined by the function $y = 6x^2 - x - 2$. When you graph this quadratic function, the curve will cross the horizontal x-axis at exactly two points: one at $x = 2/3$ and another at $x = -1/2$. These points are where the value of the function, $y$, is equal to zero, which is precisely why we call them roots or zeros. The significance of these roots extends beyond just graphing. They are fundamental in solving a wide array of problems in physics, engineering, economics, and many other fields where quadratic relationships appear. For instance, in physics, the trajectory of a projectile often follows a parabolic path, and the roots of the associated quadratic equation can tell us when the projectile hits the ground or reaches a certain height. In engineering, quadratic equations are used in designing structures, calculating stresses, and optimizing designs. Economists might use them to model cost or revenue functions and find break-even points. The fact that we found two distinct real roots for $6x^2 - x - 2 = 0$ indicates that the parabola opens upwards (since the leading coefficient, $a=6$, is positive) and its vertex lies below the x-axis, causing it to intersect the axis at two separate locations. The symmetry of the parabola is also noteworthy; the axis of symmetry is located exactly halfway between the two roots, at $x = (2/3 + (-1/2))/2 = (4/6 - 3/6)/2 = (1/6)/2 = 1/12$. This midpoint is also the x-coordinate of the vertex of the parabola. The roots are the solutions to the equation and are instrumental in understanding the behavior and properties of the quadratic function. They are the key to unlocking many practical applications where quadratic models are employed. Mastering the techniques for finding these roots equips you with a powerful analytical tool for problem-solving in various disciplines. The consistency of the results from both factoring and the quadratic formula further solidifies our confidence in the derived values and enhances our understanding of their geometric and practical implications.

Conclusion: Mastering Quadratic Equations

In conclusion, finding the roots of an equation, particularly a quadratic one like $6x^2 - x - 2 = 0$, is a fundamental skill in mathematics. We've explored two powerful methods: factoring and the quadratic formula. Factoring offers an elegant solution when applicable, breaking down the equation into simpler linear factors. The quadratic formula, on the other hand, provides a universal approach, guaranteed to yield the roots for any quadratic equation by plugging in the coefficients $a$, $b$, and $c$. Both methods successfully identified the roots as $x = 2/3$ and $x = -1/2$. These roots are not just abstract numbers; they represent critical points where the related parabola intersects the x-axis, signifying where the function's value is zero. Understanding how to find these roots is crucial for solving problems across various scientific and engineering disciplines. The ability to accurately determine the roots of quadratic equations empowers you to analyze and solve a wide range of real-world problems. Keep practicing these methods, and you'll find yourself becoming more adept at tackling algebraic challenges.

For further exploration into quadratic equations and their applications, you can visit the Khan Academy website. It offers comprehensive resources and exercises that can deepen your understanding.