Solving $7x^2 - 3x = 3$: Round To Nearest Hundredth

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Let's dive into solving the quadratic equation $7x^2 - 3x = 3$ and finding the solutions rounded to the nearest hundredth. Quadratic equations pop up frequently in mathematics, physics, and engineering, so understanding how to solve them is super useful. This article will walk you through the steps, making sure you grasp each concept along the way.

Understanding Quadratic Equations

Quadratic equations are polynomial equations of the second degree. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero. The solutions to quadratic equations, also known as roots or zeros, are the values of $x$ that satisfy the equation. These solutions can be real or complex numbers.

When we're faced with a quadratic equation, the first step is to get it into the standard form: $ax^2 + bx + c = 0$. This form makes it much easier to apply the methods for solving quadratic equations, such as factoring, completing the square, or using the quadratic formula. Identifying the coefficients $a$, $b$, and $c$ is crucial for using the quadratic formula, which is a reliable method for finding the solutions regardless of whether the equation can be easily factored. In many real-world applications, quadratic equations model parabolic trajectories, areas, or various rates of change, making them a fundamental concept in both theoretical and applied mathematics.

So, why are we even learning about quadratic equations? Well, they show up everywhere! From figuring out the trajectory of a ball you throw to designing bridges and buildings, quadratic equations are the unsung heroes behind many calculations. They help us model and understand situations where things change at a non-constant rate. So, stick with me, and let's unravel this equation together!

Rewriting the Equation

The first thing we need to do with our equation, $7x^2 - 3x = 3$, is to rewrite it in the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. This involves moving all the terms to one side of the equation, leaving zero on the other side. It's like organizing your workspace before starting a project—getting everything in the right place makes the job much smoother. To do this, we'll subtract 3 from both sides of the equation. This ensures we maintain the balance of the equation, adhering to the fundamental principle that whatever operation we perform on one side, we must perform on the other.

By subtracting 3 from both sides, we transform the original equation into $7x^2 - 3x - 3 = 0$. Now, the equation is in its standard form, making it much easier to identify the coefficients $a$, $b$, and $c$. These coefficients are crucial for using the quadratic formula, which is our next step in finding the solutions. Think of this step as setting the stage for the main act. We've cleared away the clutter and arranged the elements in a way that allows us to apply our solving tools effectively. So, with the equation now in standard form, we're well-prepared to tackle the next phase of solving for $x$.

Now we have:

$7x^2 - 3x - 3 = 0$

This form is crucial because it allows us to easily identify the coefficients $a$, $b$, and $c$, which we'll need for the quadratic formula.

Identifying Coefficients

Now that we have our equation in the standard form, $7x^2 - 3x - 3 = 0$, the next step is to pinpoint the coefficients $a$, $b$, and $c$. These coefficients are the numerical values that multiply the variables and the constant term in our quadratic equation. Think of them as the key ingredients in our formula recipe. Correctly identifying these values is crucial because they'll be directly plugged into the quadratic formula, which will help us solve for $x$.

In our equation, $7x^2 - 3x - 3 = 0$, the coefficient $a$ is the number multiplying $x^2$, which is 7. The coefficient $b$ is the number multiplying $x$, which is -3 (it's important to include the negative sign). And $c$ is the constant term, which is -3 as well. So, we have $a = 7$, $b = -3$, and $c = -3$. These values are the building blocks for the next stage, where we'll use the quadratic formula to find the solutions to our equation. Getting these coefficients right ensures that the formula will lead us to the correct answer, so it's a step we want to nail down.

So, let's break it down:

• $a$ is the coefficient of $x^2$, which is 7.
• $b$ is the coefficient of $x$, which is -3.
• $c$ is the constant term, which is -3.

These values are essential for using the quadratic formula.

Applying the Quadratic Formula

The quadratic formula is our go-to tool for solving equations in the form $ax^2 + bx + c = 0$. It's a powerful formula that provides the solutions for $x$, regardless of how complex the equation might look. The formula is:

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

This formula might look a bit intimidating at first glance, but it's really just a matter of plugging in the right numbers. Remember those coefficients $a$, $b$, and $c$ we identified earlier? This is where they come into play. The $\pm$ symbol indicates that there are potentially two solutions: one where we add the square root term and one where we subtract it. This accounts for the two possible roots of a quadratic equation.

Using the quadratic formula is like following a detailed map to find a hidden treasure. Each variable in the formula has a specific place, and by filling in the values correctly, we can navigate our way to the solutions. The formula works by considering the relationship between the coefficients and the roots of the equation, providing a systematic way to uncover those roots. So, let's take a deep breath, trust the process, and plug in our values to see what solutions we can find. With this formula in hand, we're well-equipped to solve our equation and discover the values of $x$ that make it true.

Now, we'll substitute the values of $a$, $b$, and $c$ we found earlier ($a = 7$, $b = -3$, $c = -3$) into the quadratic formula:

$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(7)(-3)}}{2(7)}$

Simplifying the Formula

Now that we've plugged our coefficients into the quadratic formula, the next step is to simplify the expression. This involves performing the arithmetic operations inside the formula, such as squaring, multiplying, and adding or subtracting. Simplifying the formula is like tidying up after a cooking session; we're organizing and clarifying the expression to make it easier to understand and work with. This process reduces the complexity and brings us closer to the actual solutions for $x$.

Let's start by simplifying the terms under the square root. We have $(-3)^2$, which equals 9, and $4(7)(-3)$, which equals -84. Since we're subtracting -84, it becomes adding 84. So, under the square root, we have $9 + 84$, which equals 93. This simplification is crucial because it condenses the expression and makes it more manageable. The part outside the square root also needs simplification. We have $-(-3)$, which becomes 3, and $2(7)$, which is 14. Now our formula looks much cleaner and clearer.

Think of this simplification as refining raw ingredients into something that's ready to be used in our final dish. We've taken the initial complex expression and broken it down into its essential parts, setting us up perfectly for the next step: calculating the two possible solutions for $x$. This methodical approach ensures accuracy and makes the solving process much more straightforward. So, with our simplified formula, we're well-prepared to find the roots of our quadratic equation.

Let's simplify step by step:

$x = \frac{3 \pm \sqrt{9 + 84}}{14}$

$x = \frac{3 \pm \sqrt{93}}{14}$

Calculating the Solutions

After simplifying the quadratic formula, we're now at the exciting stage of calculating the solutions for $x$. This is where we'll finally find the values that satisfy our original equation. Remember that $\pm$ symbol? It means we have two paths to follow, leading to two possible solutions. One path involves adding the square root term, and the other involves subtracting it. This is a key aspect of quadratic equations, as they often have two distinct roots.

To calculate these solutions, we'll first find the square root of 93, which is approximately 9.64. Now, we'll follow each path separately. For the first solution, we'll add 9.64 to 3 and then divide by 14. For the second solution, we'll subtract 9.64 from 3 and then divide by 14. These calculations will give us two values for $x$, which are the roots of our quadratic equation. Think of this step as the grand finale of our solving process, where all our previous work comes together to reveal the answers we've been seeking.

This process of finding two solutions highlights the nature of quadratic equations and their graphical representation as parabolas, which can intersect the x-axis at two points. Understanding this duality is crucial in many applications where multiple solutions can represent different scenarios or outcomes. So, with our calculator in hand, let's follow these paths and uncover the solutions that solve our equation.

We have two possible solutions:

1. $x = \frac{3 + \sqrt{93}}{14}$
2. $x = \frac{3 - \sqrt{93}}{14}$

Now, let's approximate the square root of 93:

$\sqrt{93} \approx 9.64$

So, we have:

1. $x = \frac{3 + 9.64}{14}$
2. $x = \frac{3 - 9.64}{14}$

Let's calculate each solution.

Approximating to the Nearest Hundredth

Now that we have our two potential solutions, the final step is to approximate them to the nearest hundredth. This means we'll round our answers to two decimal places, giving us a practical and easy-to-use result. Approximating to the nearest hundredth is a common practice in many real-world applications, as it provides a balance between accuracy and simplicity. It's like putting the finishing touches on a project, making sure the result is polished and ready to be used.

Let's start with the first solution, $x = \frac{3 + 9.64}{14}$. This simplifies to $x = \frac{12.64}{14}$, which is approximately 0.9028. Rounding this to the nearest hundredth gives us 0.90. For the second solution, we have $x = \frac{3 - 9.64}{14}$. This simplifies to $x = \frac{-6.64}{14}$, which is approximately -0.4743. Rounding this to the nearest hundredth gives us -0.47. So, we have our two solutions, each neatly rounded to two decimal places.

This rounding process is important because it acknowledges the practical limitations of measurement and calculation. In many real-world scenarios, we don't need infinite precision, and rounding to a sensible level of accuracy makes our results more manageable and meaningful. So, with our solutions now approximated to the nearest hundredth, we have a clear and useful answer to our quadratic equation.

Let's calculate the values and round to the nearest hundredth:

1. $x = \frac{12.64}{14} \approx 0.9028 \approx 0.90$
2. $x = \frac{-6.64}{14} \approx -0.4743 \approx -0.47$

Final Answer

After all our hard work, we've reached the final answer! We successfully solved the quadratic equation $7x^2 - 3x = 3$ and found the two solutions, approximated to the nearest hundredth. It's like reaching the summit of a challenging climb, where you can look back and appreciate the journey and the view.

Our solutions are $x \approx 0.90$ and $x \approx -0.47$. These values are the roots of the equation, meaning they are the points where the parabola represented by the equation crosses the x-axis. This final answer is not just a number; it's a testament to our understanding of quadratic equations and our ability to apply the quadratic formula effectively. We've taken a complex problem and broken it down into manageable steps, ultimately arriving at a clear and precise result.

So, congratulations on making it to the end! You've demonstrated your skill in solving quadratic equations, a valuable tool in mathematics and beyond. Remember, the process we followed here can be applied to many other quadratic equations, making this a skill that will serve you well in your mathematical journey.

Therefore, the solutions to the equation $7x^2 - 3x = 3$, rounded to the nearest hundredth, are:

$x \approx 0.90, -0.47$

In conclusion, solving quadratic equations is a fundamental skill in mathematics with wide-ranging applications. By understanding the steps involved, from rewriting the equation in standard form to applying the quadratic formula and approximating the solutions, you can confidently tackle these problems. Remember, practice makes perfect, so keep honing your skills and exploring the fascinating world of mathematics.

For further reading and to deepen your understanding of quadratic equations, visit Khan Academy's Quadratic Equations Section.