Solving $3x^2 + 12x = 7$: A Complete Guide

Let's dive into solving the quadratic equation $3x^2 + 12x = 7$ using the method of completing the square. This technique transforms a quadratic equation into a perfect square trinomial, making it easier to find the solutions. We'll break down each step to ensure clarity and understanding.

Understanding the Equation

Before we get started, let's restate our mission: We aim to find the values of $x$ that satisfy the equation $3x^2 + 12x = 7$. Quadratic equations like this pop up in various fields, from physics to engineering, so mastering the solution is super useful. The strategy we're employing here, completing the square, not only helps solve the equation but also provides insight into the structure of quadratic expressions. The beauty of this method lies in its ability to convert a standard quadratic form into a vertex form, revealing key properties of the parabola represented by the equation, such as its vertex and axis of symmetry. These features are invaluable in graphing and analyzing quadratic functions. Furthermore, understanding completing the square equips you with a foundational technique applicable to more complex problems involving conic sections and calculus. So, let's roll up our sleeves and get started, transforming our given equation into a solvable format. Remember, each step is a building block toward mastering quadratic equations, which opens doors to advanced mathematical concepts and real-world applications.

Step-by-Step Solution

Step 1: Prepare the Equation

To begin completing the square, we need the coefficient of the $x^2$ term to be 1. Currently, our equation is $3x^2 + 12x = 7$. To achieve this, we divide the entire equation by 3:

$x^2 + 4x = \frac{7}{3}$

This simplifies our equation and makes it easier to work with. Dividing by the leading coefficient ensures that we can form a perfect square trinomial more directly. This step is crucial because it aligns the equation with the standard form required for completing the square, which is $x^2 + bx + c$. By isolating the $x^2$ term, we set the stage for adding the appropriate constant to both sides of the equation, thereby creating a perfect square on the left side. This manipulation not only simplifies the algebraic process but also allows us to rewrite the quadratic expression in a form that directly reveals the vertex of the corresponding parabola. So, take a moment to appreciate this initial transformation, as it lays the groundwork for the subsequent steps that will lead us to the solutions of the equation.

Step 2: Complete the Square

Now, we complete the square on the left side of the equation. To do this, we take half of the coefficient of our $x$ term, which is 4, and square it. Half of 4 is 2, and $2^2 = 4$. We add this to both sides of the equation:

$x^2 + 4x + 4 = \frac{7}{3} + 4$

This ensures that the left side becomes a perfect square trinomial. Adding the same value to both sides maintains the equation's balance and integrity. The genius of completing the square lies in its ability to transform a general quadratic expression into a form that contains a squared term, making it much easier to solve. The value we added, derived from halving the $x$ coefficient and squaring it, is precisely what's needed to create this perfect square. Think of it as finding the missing piece of a puzzle that allows us to rewrite the equation in a more manageable form. This step is pivotal as it sets the stage for factoring the left side into a binomial squared, which will ultimately lead us to isolating $x$ and finding its values. So, take a moment to appreciate how this seemingly simple addition unlocks the potential to solve the quadratic equation.

Step 3: Simplify and Factor

Simplify the right side of the equation:

$\frac{7}{3} + 4 = \frac{7}{3} + \frac{12}{3} = \frac{19}{3}$

Now, we have:

$x^2 + 4x + 4 = \frac{19}{3}$

Factor the left side, which is now a perfect square trinomial:

$(x + 2)^2 = \frac{19}{3}$

Simplifying the right side involves basic arithmetic but is crucial for keeping the equation neat and manageable. Factoring the left side into $(x + 2)^2$ is the heart of completing the square. This step demonstrates how we've successfully transformed the original quadratic expression into a squared binomial. This form is incredibly useful because it allows us to isolate $x$ using square roots, bypassing the need for more complex methods like the quadratic formula. By recognizing the perfect square trinomial, we've made a significant leap toward solving the equation. This transformation not only simplifies the algebra but also highlights the underlying structure of the quadratic equation, revealing its relationship to a squared term. So, savor the moment as we rewrite the equation into this compact and solvable form.

Step 4: Take the Square Root

Take the square root of both sides of the equation:

$\sqrt{(x + 2)^2} = \pm \sqrt{\frac{19}{3}}$

This gives us:

$x + 2 = \pm \sqrt{\frac{19}{3}}$

Remember to include both the positive and negative square roots, as both values satisfy the equation. Taking the square root is a pivotal step as it undoes the squaring operation, bringing us closer to isolating $x$. It's important to remember that every positive number has two square roots: a positive and a negative one. Failing to consider both would lead to missing one of the solutions to the quadratic equation. The $\pm$ symbol is a reminder to account for both possibilities. This step showcases the elegance of completing the square, as it neatly unravels the equation, allowing us to solve for $x$ with relative ease. By carefully applying the square root property, we maintain mathematical rigor and ensure that we capture all possible solutions. So, embrace this step as it bridges the gap between the squared binomial and the individual values of $x$.

Step 5: Solve for x

Isolate $x$ by subtracting 2 from both sides:

$x = -2 \pm \sqrt{\frac{19}{3}}$

To rationalize the denominator, we multiply the term by $\frac{\sqrt{3}}{\sqrt{3}}$:

$x = -2 \pm \sqrt{\frac{19}{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = -2 \pm \frac{\sqrt{57}}{3}$

Thus, the solutions are:

$x = -2 + \frac{\sqrt{57}}{3}$ and $x = -2 - \frac{\sqrt{57}}{3}$

These are the exact solutions to the equation. Isolating $x$ is the final algebraic maneuver, bringing us to the culmination of our efforts. Subtracting 2 from both sides neatly separates $x$, allowing us to express its values in terms of the square root. Rationalizing the denominator is a common practice to present the solutions in a simplified and conventional form. This involves eliminating the square root from the denominator, making the expression easier to interpret and compare. The result gives us two distinct solutions, each representing a value of $x$ that satisfies the original quadratic equation. These solutions are the points where the parabola, defined by the equation, intersects the x-axis. So, celebrate this moment of triumph as we successfully unveil the solutions to the quadratic equation using the method of completing the square. Each step, from preparing the equation to isolating $x$, has contributed to this elegant and insightful result.

Conclusion

By completing the square, we found the solutions to the equation $3x^2 + 12x = 7$ to be $x = -2 + \frac{\sqrt{57}}{3}$ and $x = -2 - \frac{\sqrt{57}}{3}$. This method provides a clear and systematic way to solve quadratic equations. It also enhances our understanding of the structure and properties of quadratic expressions. Keep practicing, and you'll master this valuable algebraic technique!

For further reading and to deepen your understanding of quadratic equations, consider visiting Khan Academy's section on quadratic equations.