Solving $3x^2 = 19x - 31$: Roots In A + Bi Form

Let's dive into the world of quadratic equations and tackle the problem of finding the roots of the equation $3x^2 = 19x - 31$. Our goal is to express these roots in the simplest $a + bi$ form, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, defined as $i = \sqrt{-1}$. This exploration will not only help us solve this specific equation but also enhance our understanding of complex numbers and quadratic equations in general. So, let’s embark on this mathematical journey together!

Understanding Quadratic Equations and Complex Roots

Before we jump into solving the equation, let's establish a solid foundation. Quadratic equations are polynomial equations of the second degree, generally written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a \ne 0$. The roots of a quadratic equation are the values of $x$ that satisfy the equation. These roots can be real or complex numbers.

When the discriminant, given by the formula $D = b^2 - 4ac$, is negative, the quadratic equation has complex roots. Complex roots always come in conjugate pairs, meaning if $a + bi$ is a root, then $a - bi$ is also a root. This is a crucial concept when dealing with quadratic equations that don't have real solutions. Understanding the nature of the discriminant helps us anticipate the type of roots we'll encounter, guiding our solution process more effectively. The discriminant essentially tells us whether we're dealing with real and distinct roots, real and equal roots, or complex conjugate roots. Recognizing this at the outset can save time and prevent confusion.

Complex numbers extend the real number system by including the imaginary unit $i$, where $i^2 = -1$. A complex number is expressed in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. These numbers are essential for solving equations that have no real solutions, such as the one we're about to tackle. Grasping the properties of complex numbers, such as addition, subtraction, multiplication, and division, is key to expressing roots in the desired $a + bi$ form. The ability to manipulate complex numbers confidently allows us to simplify expressions and arrive at the most concise representation of the roots.

Transforming the Equation into Standard Form

The first step in solving our equation $3x^2 = 19x - 31$ is to rewrite it in the standard quadratic form, $ax^2 + bx + c = 0$. This form makes it easier to identify the coefficients $a$, $b$, and $c$, which are essential for applying the quadratic formula. To do this, we need to move all terms to one side of the equation, leaving zero on the other side. This process involves simple algebraic manipulation, ensuring we maintain the equation's balance by performing the same operations on both sides.

Subtracting $19x$ from both sides, we get $3x^2 - 19x = -31$. Next, adding $31$ to both sides, we arrive at the standard form: $3x^2 - 19x + 31 = 0$. Now we can clearly see that $a = 3$, $b = -19$, and $c = 31$. This transformation is a critical step because it sets the stage for using the quadratic formula, which is our primary tool for finding the roots of the equation. Correctly identifying $a$, $b$, and $c$ is paramount to ensuring we apply the quadratic formula accurately and obtain the correct solutions. This meticulous approach is fundamental to solving quadratic equations effectively.

Applying the Quadratic Formula

Now that we have the equation in standard form, we can apply the quadratic formula to find the roots. The quadratic formula is a powerful tool that provides a direct method for solving any quadratic equation. It is given by:

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

This formula is derived from the process of completing the square and is a cornerstone of algebra. It allows us to find the roots without having to factor the quadratic expression, which can be particularly useful when dealing with equations that are difficult or impossible to factor by simple methods. Understanding the derivation of the quadratic formula can also provide a deeper appreciation for its elegance and utility. By memorizing this formula and understanding how to apply it correctly, we can solve a wide range of quadratic equations with confidence.

Substituting the values $a = 3$, $b = -19$, and $c = 31$ into the quadratic formula, we get:

$x = \frac{-(-19) \pm \sqrt{(-19)^2 - 4(3)(31)}}{2(3)}$

Simplifying this expression, we have:

$x = \frac{19 \pm \sqrt{361 - 372}}{6}$

$x = \frac{19 \pm \sqrt{-11}}{6}$

Notice that we have a negative number under the square root, which indicates that the roots will be complex numbers. This is a crucial observation because it tells us that our solutions will involve the imaginary unit $i$. Recognizing this early on prepares us for working with complex numbers and ensures we express our final answers in the correct form. This step highlights the importance of understanding the discriminant and its role in determining the nature of the roots.

Expressing the Roots in $a + bi$ Form

Since we have a negative number under the square root, we can express $\sqrt{-11}$ as $\sqrt{11}i$. This is the key step in transitioning from a square root of a negative number to an expression involving the imaginary unit $i$. The imaginary unit is defined as $i = \sqrt{-1}$, so $\sqrt{-11} = \sqrt{11 \cdot -1} = \sqrt{11} \cdot \sqrt{-1} = \sqrt{11}i$. This manipulation is fundamental to working with complex numbers and allows us to write the roots in the standard $a + bi$ form.

Substituting this back into our equation, we get:

$x = \frac{19 \pm \sqrt{11}i}{6}$

Now, we can separate the real and imaginary parts to express the roots in the form $a + bi$:

$x = \frac{19}{6} \pm \frac{\sqrt{11}}{6}i$

Thus, the two roots of the equation are:

$x_1 = \frac{19}{6} + \frac{\sqrt{11}}{6}i$

$x_2 = \frac{19}{6} - \frac{\sqrt{11}}{6}i$

These roots are complex conjugates, as expected. The real part of both roots is $\frac{19}{6}$, and the imaginary parts are $\frac{\sqrt{11}}{6}$ and $-\frac{\sqrt{11}}{6}$, respectively. Expressing the roots in this form provides a clear and concise representation of the solutions, making it easy to identify the real and imaginary components. This final step underscores the importance of understanding complex numbers and their properties in solving quadratic equations.

Conclusion

In conclusion, we have successfully found the roots of the equation $3x^2 = 19x - 31$ and expressed them in the simplest $a + bi$ form. The roots are $\frac{19}{6} + \frac{\sqrt{11}}{6}i$ and $\frac{19}{6} - \frac{\sqrt{11}}{6}i$. This process involved transforming the equation into standard form, applying the quadratic formula, and simplifying the resulting expression using complex number properties. This exercise not only demonstrates the application of these mathematical tools but also reinforces our understanding of quadratic equations and complex numbers. By following these steps, we can confidently solve similar problems and deepen our mathematical expertise. Remember, practice is key, so keep exploring and solving!

For further reading and to enhance your understanding of quadratic equations and complex numbers, you might find helpful resources at Khan Academy Algebra.