Solving $x^2 + 3x - 5 = 0$ A Comprehensive Guide To Quadratic Roots
In the realm of mathematics, solving polynomial equations is a fundamental skill. Among these, quadratic equations, which take the general form of $ax^2 + bx + c = 0$, hold a significant place. This article delves into the process of finding the roots (or solutions) of a specific quadratic polynomial, $x^2 + 3x - 5$. We'll explore the quadratic formula, a powerful tool for determining the values of $x$ that satisfy the equation. The discussion will not only cover the mathematical steps but also provide a conceptual understanding of why this method works. Let’s embark on this mathematical journey to unravel the solutions to our equation.
Understanding Quadratic Equations and Roots
When we talk about quadratic equations, we're essentially dealing with polynomials of degree two. The most general form, as mentioned earlier, is $ax^2 + bx + c = 0$, where a, b, and c are constants, and a is not equal to zero. The 'roots' of a quadratic equation are the values of $x$ that make the equation true. Geometrically, these roots represent the points where the parabola (the graph of the quadratic equation) intersects the x-axis. Finding these roots is crucial in various mathematical and real-world applications, from physics to engineering and economics.
Consider our specific equation, $x^2 + 3x - 5 = 0$. Here, a = 1, b = 3, and c = -5. The challenge is to find the values of $x$ that, when plugged into this equation, will make it equal to zero. This isn't always straightforward, as simple factoring might not work. That's where the quadratic formula comes to the rescue. The quadratic formula is a universal method that provides the roots for any quadratic equation, regardless of its complexity. It's derived from the method of completing the square and is a cornerstone in algebra. Understanding its application is essential for anyone studying mathematics.
The Quadratic Formula: A Powerful Tool
The quadratic formula is a mathematical expression that provides the solutions (roots) of any quadratic equation in the standard form $ax^2 + bx + c = 0$. This formula is derived by using the method of completing the square on the general form of the quadratic equation. The formula itself is given by:
x = rac{-b inom{+}{-} ext{√}(b^2 - 4ac)}{2a}
This formula might look intimidating at first, but it's quite straightforward once you understand its components. The symbols a, b, and c correspond to the coefficients in the quadratic equation. The expression inside the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant plays a vital role in determining the nature of the roots. If the discriminant is 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 roots. The “$inom{+}{-}$” symbol indicates that there are generally two solutions, one obtained by adding the square root term and the other by subtracting it.
In the context of our equation, $x^2 + 3x - 5 = 0$, we have already identified that a = 1, b = 3, and c = -5. Now, we can plug these values into the quadratic formula to find the roots. This step-by-step process is crucial to ensure accuracy. Substituting the values correctly is the first key to solving the equation. This formula is not just a mechanical tool; it's a window into the fundamental nature of quadratic equations. It provides a way to systematically solve for the unknown, regardless of the specific numbers involved.
Applying the Quadratic Formula to $x^2 + 3x - 5 = 0$
Now, let's apply the quadratic formula to our specific equation, $x^2 + 3x - 5 = 0$. We have already established that a = 1, b = 3, and c = -5. The next step is to carefully substitute these values into the formula:
x = rac{-b inom{+}{-} ext{√}(b^2 - 4ac)}{2a}
Replacing a, b, and c with their respective values, we get:
x = rac{-3 inom{+}{-} ext{√}(3^2 - 4 * 1 * (-5))}{2 * 1}
Now, we need to simplify the expression. Start with the term inside the square root:
So, our equation now looks like this:
x = rac{-3 inom{+}{-} ext{√}29}{2}
This expression gives us two possible values for $x$: one where we add the square root of 29 and one where we subtract it. These are the two roots of the quadratic equation. The square root of 29 cannot be simplified further as 29 is a prime number, meaning it's only divisible by 1 and itself. Thus, the roots will involve the square root of 29 in their simplest form. This illustrates how the quadratic formula can lead to solutions that involve irrational numbers, which cannot be expressed as simple fractions.
Calculating the Roots: Two Distinct Solutions
From the simplified expression of the quadratic formula, $x = rac{-3 inom{+}{-} ext{√}29}{2}$, we can now determine the two distinct roots of the equation $x^2 + 3x - 5 = 0$. We have two cases to consider:
-
Adding the square root term:
x_1 = rac{-3 + ext{√}29}{2}
-
Subtracting the square root term:
x_2 = rac{-3 - ext{√}29}{2}
These two values, $x_1$ and $x_2$, are the roots of the polynomial. They represent the points where the parabola described by the equation intersects the x-axis. The fact that we have two distinct real roots is consistent with the discriminant (29) being positive. Each root is a precise value, although it involves an irrational number ( ext{√}29). This is a common occurrence when solving quadratic equations, especially when the coefficients don't lead to easily factorable expressions. Understanding that roots can be irrational is a crucial aspect of working with quadratic equations.
By calculating these roots, we have completely solved the equation. We have found the two values of $x$ that satisfy the condition $x^2 + 3x - 5 = 0$. This process demonstrates the power and utility of the quadratic formula in solving a wide range of quadratic equations.
Comparing the Calculated Roots with the Given Options
Having calculated the roots of the equation $x^2 + 3x - 5 = 0$, we now need to compare our results with the options provided in the question. Our calculated roots are:
x_1 = rac{-3 + ext{√}29}{2}
x_2 = rac{-3 - ext{√}29}{2}
Let's consider the options given:
- A. $x= rac{-3- ext{√}11}{2}$
- B. $x= rac{3- ext{√}-11}{2}$
- C. $x= rac{-3+ ext{√}11}{2}$
By comparing our calculated roots with the options, we can see that none of the options exactly match our solutions. However, it seems there might be a slight error in the provided options. Let's re-examine our calculations to ensure accuracy. We have a = 1, b = 3, and c = -5. The discriminant is $b^2 - 4ac = 3^2 - 4 * 1 * (-5) = 9 + 20 = 29$. So, the roots should indeed involve ext{√}29. The issue lies in the options presented.
Given the nature of the question, it's likely that the intended options had ext{√}29 instead of ext{√}11 or ext{√}-11. If we were to correct the options to reflect the correct discriminant, the answers would be:
- Corrected A. $x= rac{-3- ext{√}29}{2}$
- Corrected C. $x= rac{-3+ ext{√}29}{2}$
With these corrections, options A and C would match our calculated roots, $x_2$ and $x_1$, respectively. This step highlights the importance of not only knowing the method but also critically evaluating the results and given options for consistency.
Conclusion: Mastering Quadratic Equations
In conclusion, we have successfully navigated the process of finding the roots of the quadratic equation $x^2 + 3x - 5 = 0$. We began by understanding the basics of quadratic equations and the concept of roots. Then, we introduced the quadratic formula, a versatile tool for solving any quadratic equation. Applying the formula to our specific equation, we carefully substituted the coefficients and simplified the expression. This led us to the two roots:
x_1 = rac{-3 + ext{√}29}{2}
x_2 = rac{-3 - ext{√}29}{2}
During the comparison of our results with the given options, we encountered a discrepancy, which underscored the importance of critical evaluation and attention to detail. By identifying the potential error in the options, we demonstrated a comprehensive understanding of the problem-solving process.
Mastering quadratic equations is a fundamental step in mathematics. The ability to solve these equations opens doors to more advanced topics and real-world applications. The quadratic formula is a key tool in this endeavor, and understanding its application is invaluable. This journey through finding the roots of $x^2 + 3x - 5 = 0$ serves as a testament to the power and elegance of mathematical problem-solving. Remember, mathematics is not just about finding the right answer; it's about understanding the process and being able to apply that understanding in various contexts. This exploration has equipped us with the skills and insights necessary to tackle similar challenges with confidence.
