Quadratic Formula: Your Guide To Solving $2x^2 + 18 = 4x$

When you're faced with a quadratic equation, especially one that looks a bit jumbled like $2x^2 + 18 = 4x$, the quadratic formula often comes to the rescue. It's a powerful tool that allows you to find the roots (or solutions) of any quadratic equation in the standard form $ax^2 + bx + c = 0$. Let's dive into how we can use this formula to solve our specific equation and break down each step to make it crystal clear. The beauty of the quadratic formula is its universality; it works for all quadratic equations, whether they have real roots, complex roots, or even a single repeated root. It's like a master key that unlocks the solutions to a vast array of algebraic problems. So, get ready to demystify this essential mathematical concept and apply it directly to conquer the equation $2x^2 + 18 = 4x$. We'll start by rearranging the equation into the standard form, identify the coefficients $a$, $b$, and $c$, and then plug them into the formula. Throughout this process, we'll emphasize simplification techniques to ensure the roots are presented in their simplest form, a crucial aspect when dealing with square roots and fractions. Understanding the quadratic formula isn't just about memorizing an equation; it's about grasping the underlying principles of algebra and developing problem-solving skills that extend far beyond this particular problem. It’s a cornerstone of algebra that forms the basis for many advanced mathematical concepts.

Understanding the Standard Form and Identifying Coefficients

Before we can wield the quadratic formula, we must first ensure our equation is in the standard form: $ax^2 + bx + c = 0$. Our given equation is $2x^2 + 18 = 4x$. To get it into the standard form, we need to move all terms to one side, typically the left side, so that the equation equals zero. We can achieve this by subtracting $4x$ from both sides of the equation: $2x^2 - 4x + 18 = 0$. Now, our equation is neatly arranged in the standard quadratic form. From here, we can easily identify the coefficients: $a$, $b$, and $c$. The coefficient $a$ is the number multiplying the $x^2$ term, which is $2$ in our case. The coefficient $b$ is the number multiplying the $x$ term, which is $-4$ (don't forget the negative sign!). Finally, the coefficient $c$ is the constant term, which is $18$. It is absolutely critical to correctly identify these coefficients, as any mistake here will lead to an incorrect solution. Take your time with this step, double-checking each value against the standard form. Sometimes, equations might be presented with terms out of order, or even involve fractions or decimals, making this initial rearrangement and identification process even more important. Remember, the goal is to isolate the terms in descending order of their powers of $x$, ending with the constant term, all set equal to zero. This structured approach ensures consistency and reduces the chances of errors as we move forward. For instance, if the equation was $-3x + 5x^2 - 7 = 0$, you would rearrange it to $5x^2 - 3x - 7 = 0$, identifying $a=5$, $b=-3$, and $c=-7$. This foundational step is where the journey to finding the roots truly begins, setting the stage for the application of the quadratic formula.

Applying the Quadratic Formula: Step-by-Step

With our equation $2x^2 - 4x + 18 = 0$ in standard form and our coefficients identified ($a=2$, $b=-4$, $c=18$), we are ready to apply the quadratic formula. The formula itself is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This formula might look a bit intimidating at first, but let's break it down piece by piece. First, we substitute the values of $a$, $b$, and $c$ into the formula. So, we get: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(18)}}{2(2)}$. Let's simplify each part of this expression. The term $-b$ becomes $-(-4)$, which simplifies to $4$. The term $b^2$ becomes $(-4)^2$, which is $16$. The term $4ac$ becomes $4(2)(18)$, which multiplies out to $144$. The denominator $2a$ becomes $2(2)$, which is $4$. Now, let's substitute these simplified values back into the formula: $x = \frac{4 \pm \sqrt{16 - 144}}{4}$. The next crucial step involves simplifying the expression under the square root, which is known as the discriminant ($b^2 - 4ac$). In our case, $16 - 144 = -128$. So, the formula now looks like: $x = \frac{4 \pm \sqrt{-128}}{4}$. This is where we encounter a negative number under the square root. In the realm of real numbers, the square root of a negative number is undefined. However, in the realm of complex numbers, we can proceed. The square root of $-128$ can be written as $\sqrt{128}i$, where $i$ is the imaginary unit, defined as $\sqrt{-1}$. To simplify $\sqrt{128}$, we look for the largest perfect square that divides $128$. That perfect square is $64$, since $64 \times 2 = 128$. Therefore, $\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2}$. So, $\sqrt{-128} = 8i\sqrt{2}$. Substituting this back into our equation for $x$: $x = \frac{4 \pm 8i\sqrt{2}}{4}$. This is the expression for our roots. The final step in this section is to simplify this fraction. We can divide each term in the numerator by the denominator. So, $x = \frac{4}{4} \pm \frac{8i\sqrt{2}}{4}$. This simplifies to $x = 1 \pm 2i\sqrt{2}$. This gives us two complex roots: $x_1 = 1 + 2i\sqrt{2}$ and $x_2 = 1 - 2i\sqrt{2}$. The process of substitution and simplification is the heart of using the quadratic formula effectively. Each mathematical operation must be performed with care, especially when dealing with signs and radicals. The discriminant, in particular, is a vital part of the formula as it tells us about the nature of the roots (real, complex, or repeated). When the discriminant is negative, as it is here, we know we'll be working with complex conjugate roots. The ability to simplify radicals is also key to presenting the solution in its simplest form.

Simplifying the Roots to Their Simplest Form

One of the most important aspects of solving equations using the quadratic formula is presenting the roots in their simplest form. This involves two main aspects: simplifying any radicals and simplifying any fractions. We've already touched upon simplifying the radical $\sqrt{-128}$, but let's revisit it to ensure clarity. We found that $\sqrt{-128} = 8i\sqrt{2}$. The number $128$ was broken down into its prime factors: $128 = 2^7$. To simplify the square root, we look for pairs of prime factors. We have three pairs of $2$s, leaving one $2$ remaining. Each pair of $2$s contributes a $2$ outside the radical. So, $\sqrt{128} = 2 \times 2 \times 2 \sqrt{2} = 8\sqrt{2}$. The presence of the negative sign under the square root introduces the imaginary unit $i$, where $i = \sqrt{-1}$. Thus, $\sqrt{-128} = \sqrt{-1} \times \sqrt{128} = i \times 8\sqrt{2} = 8i\sqrt{2}$. This is the simplest form of the radical part. The next step in simplification is to simplify the overall fraction. Our expression for $x$ was $x = \frac{4 \pm 8i\sqrt{2}}{4}$. To simplify this fraction, we divide each term in the numerator by the common denominator, which is $4$. So, we get $x = \frac{4}{4} \pm \frac{8i\sqrt{2}}{4}$. Performing the division for each term: $\frac{4}{4} = 1$ and $\frac{8i\sqrt{2}}{4} = 2i\sqrt{2}$. Combining these, we arrive at the simplest form of the roots: $x = 1 \pm 2i\sqrt{2}$. This means our two roots are $x_1 = 1 + 2i\sqrt{2}$ and $x_2 = 1 - 2i\sqrt{2}$. These are complex conjugate roots, which is expected when the discriminant ($b^2 - 4ac$) is negative. Simplifying fractions involving radicals and imaginary numbers requires careful attention to detail. Always look for common factors between the numerator and the denominator. In cases with multiple terms in the numerator, each term must be divided by the denominator. The goal is to reduce the numbers as much as possible, ensuring that no further simplification is possible. For instance, if we had $\frac{6 \pm \sqrt{12}}{2}$, we would first simplify $\sqrt{12}$ to $2\sqrt{3}$, giving us $\frac{6 \pm 2\sqrt{3}}{2}$. Then, we divide each term by $2$ to get $3 \pm \sqrt{3}$. This is the simplest form. Mastering the art of simplification ensures that your answers are not only correct but also presented in the most concise and understandable manner, which is a hallmark of strong mathematical proficiency. It's a skill that is honed with practice and attention to the fundamental rules of arithmetic and algebra.

Conclusion: The Power of the Quadratic Formula

We've successfully navigated the process of solving the quadratic equation $2x^2 + 18 = 4x$ using the quadratic formula. By first rearranging the equation into the standard form $ax^2 + bx + c = 0$, we identified $a=2$, $b=-4$, and $c=18$. Plugging these values into the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, we performed detailed calculations. The discriminant, $b^2 - 4ac$, evaluated to $-128$, indicating that the roots would be complex. We then simplified the square root of $-128$ to $8i\sqrt{2}$. Finally, after substituting this back into the formula and simplifying the resulting fraction, we arrived at the roots in their simplest form: $x = 1 \pm 2i\sqrt{2}$. This means the two solutions are $x_1 = 1 + 2i\sqrt{2}$ and $x_2 = 1 - 2i\sqrt{2}$. The quadratic formula is an indispensable tool in algebra, providing a reliable method for finding the roots of any quadratic equation. Its power lies in its consistency and its ability to handle all types of quadratic equations, including those with complex roots, which arise when the discriminant is negative. Understanding how to simplify radicals and fractions is crucial for presenting these solutions in their most elegant and understandable form. This skill is not just about getting the right answer; it's about demonstrating a thorough grasp of mathematical principles. Keep practicing, and you'll find that applying the quadratic formula becomes second nature, opening doors to solving more complex mathematical challenges. The ability to solve quadratic equations is a fundamental skill in mathematics, with applications in various fields, including physics, engineering, economics, and computer science. It is a building block for understanding more advanced mathematical concepts and problem-solving techniques. For further exploration into quadratic equations and other algebraic concepts, you can refer to resources like Khan Academy or Wolfram MathWorld.