Solving Quadratic Equation $4x^2 + 45x + 24 = 0$ Find The Other Solution

Hey guys! Let's dive into the fascinating world of quadratic equations. Today, we're going to tackle the equation $4x^2 + 45x + 24 = 0$. We already know one of the solutions, thanks to our trusty quadratic formula, but the quest isn't over yet! We need to find that elusive second solution, rounding our answer to the nearest hundredth.

Understanding Quadratic Equations and the Quadratic Formula

Before we jump into solving, let's quickly recap what we're dealing with. Quadratic equations are polynomial equations of the second degree, meaning the highest power of the variable (in this case, x) is 2. They generally take the form $ax^2 + bx + c = 0$, where a, b, and c are constants. In our equation, we have a = 4, b = 45, and c = 24.

The quadratic formula is our superhero when it comes to solving these equations. It provides a direct way to find the roots (or solutions) of any quadratic equation. The formula looks like this:

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

This formula might seem a bit intimidating at first, but it's actually quite straightforward once you get the hang of it. The "±" symbol indicates that there are two possible solutions, one obtained by adding the square root term and the other by subtracting it. This is because quadratic equations, due to their squared term, often have two distinct roots.

Now, let's break down the components of the quadratic formula and see how they apply to our specific equation. The term inside the square root, $b^2 - 4ac$, is called the discriminant. The discriminant is super important because it tells us about the nature of the roots. If the discriminant is positive, we have two distinct real roots (which is the case for our equation since we already know one solution is -10.69). If it's zero, we have one real root (a repeated root). And if it's negative, we have two complex roots. Understanding the discriminant can give you a quick insight into the solutions you expect to find.

In our case, let's calculate the discriminant: $45^2 - 4 * 4 * 24 = 2025 - 384 = 1641$. Since 1641 is positive, we confirm that we should indeed have two distinct real roots. We already know one of them, so we're on the right track to finding the other!

Plugging in the Values and Finding the Solutions

Alright, with the quadratic formula in our arsenal and the values of a, b, and c identified, let's plug them in and see what we get. Remember, our equation is $4x^2 + 45x + 24 = 0$, so a = 4, b = 45, and c = 24. Substituting these values into the quadratic formula gives us:

$x = \frac{-45 \pm \sqrt{45^2 - 4 * 4 * 24}}{2 * 4}$

Simplifying the expression under the square root, as we calculated earlier, we get:

$x = \frac{-45 \pm \sqrt{1641}}{8}$

Now, we need to calculate the square root of 1641. It's not a perfect square, but using a calculator, we find that $\sqrt{1641} \approx 40.51$. So, our equation becomes:

$x = \frac{-45 \pm 40.51}{8}$

This gives us two possible solutions:

1. $x_1 = \frac{-45 + 40.51}{8} \approx \frac{-4.49}{8} \approx -0.56$
2. $x_2 = \frac{-45 - 40.51}{8} \approx \frac{-85.51}{8} \approx -10.69$

Lo and behold, we already know that one solution is approximately -10.69, which matches our second calculation. That means the other solution, rounded to the nearest hundredth, is approximately -0.56. Awesome!

Verifying the Solution and Understanding the Implications

It's always a good idea to double-check our work, right? We can do this by plugging our solutions back into the original equation and seeing if they make the equation true. Let's plug -0.56 into $4x^2 + 45x + 24 = 0$:

$4(-0.56)^2 + 45(-0.56) + 24 \approx 4(0.31) - 25.2 + 24 \approx 1.24 - 25.2 + 24 \approx 0.04$

This is pretty close to zero, which confirms that -0.56 is indeed a solution (remember, we rounded our answer, so there might be a tiny bit of error). If we plugged in -10.69, we'd get a similar result, further solidifying our solutions.

But beyond just getting the right answer, it's important to understand what these solutions mean. The solutions to a quadratic equation represent the points where the parabola defined by the equation intersects the x-axis. Think of the graph of $y = 4x^2 + 45x + 24$. It's a U-shaped curve (a parabola), and the points where it crosses the x-axis (where y = 0) are at x = -0.56 and x = -10.69. This visual representation can help you grasp the concept of roots in a more intuitive way.

Alternative Methods: Factoring (When Possible)

While the quadratic formula is a foolproof method for solving quadratic equations, there's another technique called factoring that can be quicker and easier in certain cases. Factoring involves rewriting the quadratic expression as a product of two binomials. For example, if we could factor $4x^2 + 45x + 24$ into something like $(Ax + B)(Cx + D)$, then we could find the roots by setting each binomial equal to zero and solving for x. However, in this particular case, factoring isn't straightforward due to the coefficients and the lack of easily identifiable factors. So, the quadratic formula is definitely the way to go here.

Common Mistakes to Avoid

Before we wrap up, let's touch on some common pitfalls that people often encounter when using the quadratic formula. One frequent mistake is messing up the signs, especially with the "-b" term and the "-4ac" term. Double-check those signs! Another common error is incorrectly simplifying the expression under the square root or forgetting to divide the entire numerator by 2a. Always take your time and carefully follow the order of operations (PEMDAS/BODMAS). And of course, a calculator is your friend for those square roots and arithmetic calculations, but make sure you're entering the numbers correctly!

Conclusion: Mastering the Quadratic Formula

So there you have it, guys! We successfully solved the quadratic equation $4x^2 + 45x + 24 = 0$ using the quadratic formula and found the other solution to be approximately -0.56. We also took a deeper dive into understanding quadratic equations, the discriminant, and the implications of the solutions. Remember, the quadratic formula is a powerful tool in your mathematical arsenal. With practice and a clear understanding of the concepts, you'll be able to tackle any quadratic equation that comes your way!

Keep practicing, keep exploring, and most importantly, keep having fun with math!