Solving Quadratic Equations: A Step-by-Step Guide

Hey guys! Today, we're going to tackle a quadratic equation using the quadratic formula. Specifically, we’ll solve the equation $8m^2 - 2m = 11$. Don't worry; it's not as intimidating as it looks! Let's break it down step by step.

1. Understanding Quadratic Equations

Before diving into the quadratic formula, let's quickly recap what a quadratic equation is. A quadratic equation is an equation of the form:

$ax^2 + bx + c = 0$

where a, b, and c are constants, and 'x' is the variable we want to solve for. The quadratic formula is a neat tool that helps us find the values of 'x' that satisfy this equation. It's especially handy when factoring isn't straightforward.

So, what makes this type of equation so special? Well, quadratic equations pop up all over the place in real-world applications. Think about the trajectory of a ball thrown in the air, the design of curved architectural structures, or even optimizing business processes. Understanding how to solve them opens up a whole new world of problem-solving possibilities.

Why is it called 'quadratic,' anyway? The term 'quadratic' comes from the Latin word 'quadratus,' which means square. This refers to the fact that the highest power of the variable in the equation is 2 (i.e., $x^2$). This little piece of trivia can help you remember the key characteristic of quadratic equations!

Now, let's talk about why the quadratic formula is such a big deal. Sure, you can sometimes solve quadratic equations by factoring, but that's not always easy or even possible. The quadratic formula, on the other hand, works every single time, no matter how messy the coefficients are. It's like having a universal key that unlocks any quadratic equation you throw at it.

But the quadratic formula isn't just a magic bullet; it also gives us valuable insights into the nature of the solutions. The discriminant (the part under the square root) tells us whether we'll have two real solutions, one real solution, or two complex solutions. This information can be crucial in understanding the behavior of the system the equation represents.

2. Preparing the Equation

Our given equation is $8m^2 - 2m = 11$. The first thing we need to do is rewrite it in the standard form $ax^2 + bx + c = 0$. To do this, we subtract 11 from both sides:

$8m^2 - 2m - 11 = 0$

Now, we can identify our coefficients:

• $a = 8$
• $b = -2$
• $c = -11$

Making sure your equation is in the standard form is a crucial first step. It ensures that you correctly identify the coefficients a, b, and c, which are essential for plugging into the quadratic formula. A simple mistake here can throw off your entire solution, so take your time and double-check your work.

Why is this standard form so important? Well, it's all about consistency. By agreeing on a standard way to write quadratic equations, mathematicians have made it easier to communicate and work with these equations. It's like having a common language that everyone understands. This standardization simplifies the process of applying the quadratic formula and interpreting the results.

Also, remember that the order of the terms matters. The $ax^2$ term always comes first, followed by the $bx$ term, and finally the constant term $c$. This order ensures that you correctly identify the coefficients and avoid mixing them up.

Another common mistake is forgetting to include the sign of the coefficients. For example, if your equation is $x^2 - 5x + 6 = 0$, then a = 1, b = -5, and c = 6. Pay close attention to the negative signs, as they can significantly impact your final answer.

3. Applying the Quadratic Formula

The quadratic formula is:

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

Now, plug in the values of a, b, and c we identified earlier:

$m = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(8)(-11)}}{2(8)}$

Simplify step by step:

$m = \frac{2 \pm \sqrt{4 + 352}}{16}$

$m = \frac{2 \pm \sqrt{356}}{16}$

Don't be intimidated by the square root! It's just a number, and we'll deal with it shortly. The key here is to take your time and be meticulous with your calculations. A single mistake in arithmetic can lead to a completely wrong answer.

When plugging the values into the formula, pay special attention to the signs. Remember that -(-2) becomes +2. Also, be careful when squaring negative numbers, as (-2)^2 is positive 4.

The $\pm$ symbol in the quadratic formula means that we'll have two possible solutions: one where we add the square root and one where we subtract it. This is because quadratic equations typically have two roots (solutions). These roots represent the points where the parabola defined by the equation intersects the x-axis.

Another important thing to remember is the order of operations (PEMDAS/BODMAS). Make sure you perform the operations inside the square root first, then multiply and divide before adding or subtracting.

4. Simplifying the Solution

We have $m = \frac{2 \pm \sqrt{356}}{16}$. We can simplify $\sqrt{356}$ by factoring out the largest perfect square. $356 = 4 * 89$, so $\sqrt{356} = \sqrt{4 * 89} = 2\sqrt{89}$.

Now our equation looks like this:

$m = \frac{2 \pm 2\sqrt{89}}{16}$

We can factor out a 2 from the numerator:

$m = \frac{2(1 \pm \sqrt{89})}{16}$

And simplify by dividing both numerator and denominator by 2:

$m = \frac{1 \pm \sqrt{89}}{8}$

So, our two solutions are:

$m = \frac{1 + \sqrt{89}}{8}$ and $m = \frac{1 - \sqrt{89}}{8}$

Simplifying radicals can sometimes be tricky, but it's a valuable skill to have. The goal is to find the largest perfect square that divides evenly into the number under the radical. In this case, we found that 4 is the largest perfect square that divides into 356.

Why bother simplifying radicals at all? Well, it makes the solutions easier to work with and compare. It also helps to express the solutions in their simplest form, which is often preferred in mathematical contexts.

If you're not comfortable simplifying radicals by hand, you can always use a calculator to approximate the value of the square root. However, keep in mind that this will give you a decimal approximation, rather than the exact solution in radical form.

Another important thing to remember is that you can only simplify radicals if the number under the radical has a perfect square factor. If it doesn't, then the radical is already in its simplest form.

5. Final Solutions

Therefore, the solutions to the equation $8m^2 - 2m = 11$ are:

$m = \frac{1 + \sqrt{89}}{8} \approx 1.29$ and $m = \frac{1 - \sqrt{89}}{8} \approx -1.04$

These are the two values of 'm' that satisfy the original equation. You can plug them back into the equation to verify that they work!

Congratulations! You've successfully solved a quadratic equation using the quadratic formula. Remember, practice makes perfect, so keep working on different examples to build your skills.

Now that we've found the solutions, let's take a moment to reflect on what they mean. In the context of a quadratic equation, the solutions (also called roots or zeros) represent the points where the parabola defined by the equation intersects the x-axis.

If the discriminant (the part under the square root) is positive, as it is in this case, then the equation has two distinct real solutions. This means that the parabola intersects the x-axis at two different points.

If the discriminant is zero, then the equation has one real solution (a repeated root). This means that the parabola touches the x-axis at exactly one point.

If the discriminant is negative, then the equation has two complex solutions. This means that the parabola does not intersect the x-axis at all.

Understanding the nature of the solutions can give you valuable insights into the behavior of the system that the equation represents.

So there you have it! Solving quadratic equations using the quadratic formula might seem daunting at first, but with a little practice and a step-by-step approach, you can master it. Keep practicing, and you'll be solving quadratic equations like a pro in no time!