Solving Quadratic Equations: Factoring And Decimal Solutions

Hey guys! Let's dive into the world of quadratic equations! We're going to tackle two problems today. First, we'll solve $3x^2 + 2x - 8 = 0$ by factoring. Then, we'll solve $2x^2 + 7x - 8 = 0$ and round our answers to two decimal places. Get your thinking caps on, because this is going to be fun!

Solving $3x^2 + 2x - 8 = 0$ by Factoring

Let's kick things off by understanding the importance of factoring in solving quadratic equations. Factoring, in essence, is the art of breaking down a complex expression into simpler components—think of it like dismantling a machine to understand its workings. In the context of quadratics, we aim to rewrite the equation in a form that reveals its roots (or solutions) more clearly. This method hinges on the zero-product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This is a cornerstone principle that allows us to transition from a factored form to individual solutions. The beauty of factoring lies in its elegance and efficiency when applied to suitable quadratic equations. It's a direct route to the solutions, avoiding the sometimes cumbersome process of using the quadratic formula. However, it's worth noting that not all quadratic equations are factorable over integers, which leads us to explore other methods when factoring isn't an option. For those equations that do yield to factoring, it offers a satisfying and insightful way to uncover the roots. So, mastering this technique is not just about finding answers; it's about deepening our understanding of the structure and properties of quadratic equations.

Our first task is to solve the quadratic equation $3x^2 + 2x - 8 = 0$ by factoring. Factoring is a method used to simplify equations by expressing them as a product of two or more factors. This is especially useful for quadratic equations because if we can factor the quadratic expression, we can then use the zero-product property to find the solutions. Remember, the zero-product property states that if the product of two or more factors is zero, then at least one of the factors must be zero.

To factor the quadratic $3x^2 + 2x - 8$, we're looking for two binomials that, when multiplied, give us the original quadratic. This involves a bit of trial and error, but there's a systematic way to approach it. First, we need to find two numbers that multiply to the product of the leading coefficient (3) and the constant term (-8), which is -24. At the same time, these two numbers must add up to the middle coefficient, which is 2. After considering different factor pairs of -24, we find that the numbers 6 and -4 satisfy these conditions because 6 * -4 = -24 and 6 + (-4) = 2.

Now, we rewrite the middle term (2x) using these numbers. The equation becomes $3x^2 + 6x - 4x - 8 = 0$. Notice how we've split the 2x term into 6x and -4x. This step is crucial for factoring by grouping, which is our next move. We group the first two terms and the last two terms together: $(3x^2 + 6x) + (-4x - 8) = 0$. Now, we factor out the greatest common factor (GCF) from each group. From the first group, the GCF is 3x, and from the second group, it's -4. Factoring these out, we get $3x(x + 2) - 4(x + 2) = 0$.

Observe that we now have a common binomial factor of $(x + 2)$ in both terms. We factor this out to get $(3x - 4)(x + 2) = 0$. This is the factored form of our original quadratic equation. We've successfully transformed the equation into a product of two binomials, which is the key to using the zero-product property. Now, we set each factor equal to zero and solve for x. This gives us two separate equations: $3x - 4 = 0$ and $x + 2 = 0$.

Solving $3x - 4 = 0$, we add 4 to both sides to get $3x = 4$, and then divide by 3 to find x = rac{4}{3}. For the second equation, $x + 2 = 0$, we simply subtract 2 from both sides to get $x = -2$. So, the solutions to the quadratic equation $3x^2 + 2x - 8 = 0$ are x = rac{4}{3} and $x = -2$. These are the values of x that make the equation true, and we found them by the elegant method of factoring! This whole process underscores the power of factoring as a technique for solving quadratic equations, especially when the roots are rational numbers.

Solving $2x^2 + 7x - 8 = 0$ and Rounding to Two Decimal Places

Now, let's move on to the second equation, $2x^2 + 7x - 8 = 0$. This one's a little trickier because it doesn't factor nicely using integers. That's where the quadratic formula comes to the rescue! The quadratic formula is a universal tool for solving any quadratic equation, regardless of whether it can be factored easily or not. It's derived from the process of completing the square, and it provides a direct method for finding the roots of any quadratic equation in the standard form $ax^2 + bx + c = 0$. Understanding the quadratic formula is crucial because it bridges the gap when factoring isn't feasible. It's not just a formula to memorize; it's a testament to the power of algebraic manipulation and the completeness of our mathematical toolkit for solving quadratic equations. Whether the roots are real or complex, rational or irrational, the quadratic formula provides a reliable pathway to their discovery.

The quadratic formula is a powerful tool that gives us the solutions to any quadratic equation in the form $ax^2 + bx + c = 0$. The formula is given by:

x = rac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In our equation, $2x^2 + 7x - 8 = 0$, we can identify $a = 2$, $b = 7$, and $c = -8$. These coefficients are the key ingredients we need to plug into the quadratic formula. It's like having the recipe for a cake; we just need to add the ingredients in the right amounts. The coefficients dictate the shape and position of the parabola that the quadratic equation represents, and the quadratic formula uses these coefficients to pinpoint where the parabola intersects the x-axis—these intersection points are the solutions to the equation. By carefully identifying a, b, and c, we set the stage for a straightforward application of the formula, leading us to the roots of the equation. It's a methodical process that transforms a potentially daunting problem into a manageable calculation.

Let's substitute these values into the quadratic formula:

x = rac{-7 \pm \sqrt{7^2 - 4(2)(-8)}}{2(2)}

Now, we simplify step by step. First, we calculate the value inside the square root. We have $7^2$ which is 49, and then we have $-4(2)(-8)$, which is +64. Adding these together, we get $49 + 64 = 113$. So, the expression under the square root simplifies to 113. This number, the discriminant, is particularly interesting because it tells us about the nature of the roots. A positive discriminant, like ours, indicates that we have two distinct real roots. If the discriminant were zero, we'd have one real root (a repeated root), and if it were negative, we'd have two complex roots. For now, a positive discriminant means we're on the right track to finding two real solutions. This little detour into the significance of the discriminant highlights the depth of information contained within the quadratic formula, making it not just a tool for finding solutions, but also a window into the behavior of quadratic equations.

Our equation now looks like this:

x = rac{-7 \pm \sqrt{113}}{4}

Since 113 is not a perfect square, we'll need to approximate the square root of 113. Using a calculator, we find that $\sqrt{113} \approx 10.63$. The next step is to deal with the $\pm$ sign, which means we actually have two separate equations to calculate. This is a crucial aspect of the quadratic formula, representing the two potential points where the parabola crosses the x-axis. One equation will use the plus sign, giving us one solution, and the other will use the minus sign, leading to the second solution. This dual nature is a direct consequence of the parabolic shape of quadratic equations, which can intersect the x-axis at two points, one point, or not at all. By splitting the equation at this stage, we ensure that we capture both solutions (if they exist), providing a complete picture of the equation's roots.

So, we have two possible solutions:

x_1 = rac{-7 + 10.63}{4}

x_2 = rac{-7 - 10.63}{4}

Let's calculate each of these. For $x_1$, we have $(-7 + 10.63)$ which equals 3.63. Dividing this by 4 gives us approximately 0.9075. Remember, we need to round our answers to two decimal places, so $x_1 \approx 0.91$. Now, let's tackle $x_2$. Here, we have $(-7 - 10.63)$ which is -17.63. Dividing this by 4 gives us approximately -4.4075. Again, rounding to two decimal places, we get $x_2 \approx -4.41$. These rounded values are our final solutions, carefully adjusted to meet the precision requirement. The process of rounding, while seemingly simple, is a critical step in applied mathematics, ensuring that our answers are both accurate and practical for real-world applications.

Therefore:

$x_1 \approx 0.91$

$x_2 \approx -4.41$

So, the solutions to the equation $2x^2 + 7x - 8 = 0$, correct to two decimal places, are approximately 0.91 and -4.41. We've successfully navigated this quadratic equation using the quadratic formula, a testament to its power and versatility. These solutions represent the x-values where the parabola defined by the equation intersects the x-axis. They are the roots of the equation, the values that make the equation true. The journey from the initial equation to these final solutions highlights the beauty of algebra: transforming complex expressions into understandable and actionable results.

Conclusion

Alright guys, we've done it! We solved $3x^2 + 2x - 8 = 0$ by factoring and found the solutions to be x = rac{4}{3} and $x = -2$. Then, we tackled $2x^2 + 7x - 8 = 0$ using the quadratic formula and rounded our answers to two decimal places, getting approximately 0.91 and -4.41. Whether it's the elegance of factoring or the power of the quadratic formula, we've shown that we can conquer any quadratic equation that comes our way. Keep practicing, and you'll be quadratic equation masters in no time!