Values Of C And D For I(2+3i)(c+di) To Be A Real Number

Hey everyone! Today, we're diving into a fun little problem from the realm of complex numbers. We've got this expression: $i(2+3i)(c+di)$, and our mission, should we choose to accept it, is to figure out what values of $c$ and $d$ will make this whole thing a real number. Buckle up, because we're about to embark on a mathematical adventure!

Understanding the Problem

Before we jump into solving, let's break down what we're dealing with. The expression involves complex numbers, which are numbers that can be written in the form $a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i$ is the imaginary unit (defined as the square root of -1). Our goal is to find values for $c$ and $d$ (which are also real numbers) such that when we simplify the expression, the imaginary part vanishes, leaving us with only a real number. In simpler terms, we want to get rid of any $i$ terms in our final answer. This means the coefficient of $i$ must be zero.

So, the core question we're tackling is: What constraints must we impose on $c$ and $d$ so that the expression $i(2+3i)(c+di)$ results in a real number? We'll need to use the properties of complex numbers, such as the distributive property and the fact that $i^2 = -1$, to simplify the expression and isolate the real and imaginary parts. This involves expanding the product, combining like terms, and then setting the imaginary part equal to zero. By solving for the conditions on $c$ and $d$ that make the imaginary part zero, we'll find the values that make the original expression a real number.

Step-by-Step Simplification

First, let's tackle the product $(2+3i)(c+di)$. We'll use the good ol' distributive property (also known as the FOIL method) to expand this:

$(2+3i)(c+di) = 2(c+di) + 3i(c+di) $

$= 2c + 2di + 3ci + 3di^2 $

Now, remember that $i^2 = -1$, so we can substitute that in:

$= 2c + 2di + 3ci - 3d$

Let's group the real and imaginary parts together:

$= (2c - 3d) + (2d + 3c)i$

Great! Now we have the expression in the form $a + bi$, where $a = 2c - 3d$ and $b = 2d + 3c$. But we're not done yet! We still need to multiply this by $i$:

$i[(2c - 3d) + (2d + 3c)i] = i(2c - 3d) + i(2d + 3c)i $

$= (2c - 3d)i + (2d + 3c)i^2$

Again, $i^2 = -1$, so:

$= (2c - 3d)i - (2d + 3c)$

Now, let's rearrange it to the standard $a + bi$ form:

$= -(2d + 3c) + (2c - 3d)i$

Isolating the Real and Imaginary Parts

Okay, so after all that algebraic maneuvering, we've arrived at a crucial point. Our expression $i(2+3i)(c+di)$ is now simplified to $-(2d + 3c) + (2c - 3d)i$. Remember, we want this expression to be a real number. What does that mean in terms of its real and imaginary parts? It means the imaginary part must be equal to zero. In other words, the coefficient of $i$ must be zero. So, we need to solve the equation $2c - 3d = 0$.

The real part of our complex number is $-(2d + 3c)$. For the expression to be a real number, the imaginary part must be zero. Thus, we focus on the imaginary part which is given by $(2c - 3d)i$. Setting the coefficient of $i$ to zero gives us the equation:

$2c - 3d = 0$

This is a linear equation that relates $c$ and $d$. To find the values of $c$ and $d$ that satisfy this condition, we need to consider the implications of this equation. Essentially, it tells us that $2c$ must be equal to $3d$. We can rewrite this equation in several ways to get a better understanding of the relationship between $c$ and $d$.

Finding the Conditions for a Real Number

We've established that for $i(2+3i)(c+di)$ to be a real number, the imaginary part of the simplified expression must be zero. This led us to the equation $2c - 3d = 0$. Let's explore this equation further to understand the relationship between $c$ and $d$.

We can rewrite the equation as:

$2c = 3d$

From this, we can express $c$ in terms of $d$ or vice versa. Let's express $c$ in terms of $d$:

c = rac{3}{2}d

This equation tells us that the value of $c$ must be rac{3}{2} times the value of $d$ for the expression to be a real number. This gives us a condition that $c$ and $d$ must satisfy. We can also express $d$ in terms of $c$:

d = rac{2}{3}c

This tells us that the value of $d$ must be rac{2}{3} times the value of $c$. Both expressions are equivalent and give us the same condition: $2c = 3d$.

The key takeaway here is that there isn't a single unique pair of values for $c$ and $d$ that will make the expression a real number. Instead, there are infinitely many pairs of values that satisfy the condition $2c = 3d$. For example, if we choose $d = 2$, then c = rac{3}{2}(2) = 3. If we choose $c = 6$, then d = rac{2}{3}(6) = 4. So, the pairs $(3, 2)$ and $(6, 4)$ are just two examples of infinitely many pairs that will work.

Examples and Scenarios

To solidify our understanding, let's explore a few examples. We know that the condition $2c = 3d$ must be satisfied for the expression $i(2+3i)(c+di)$ to be a real number.

• Scenario 1: Let's pick d = 2.
  Using our condition $2c = 3d$, we substitute $d = 2$ to find $c$: $2c = 3(2)$ $2c = 6$ $c = 3$ So, when $c = 3$ and $d = 2$, the expression should be a real number. Let's verify: $i(2+3i)(3+2i) = i(6 + 4i + 9i + 6i^2) = i(6 + 13i - 6) = i(13i) = 13i^2 = -13$, which is indeed a real number. This confirms that our condition works.

• Scenario 2: Let's pick c = -3.
  Using our condition $2c = 3d$, we substitute $c = -3$ to find $d$: $2(-3) = 3d$ $-6 = 3d$ $d = -2$ So, when $c = -3$ and $d = -2$, the expression should be a real number. Let's check: $i(2+3i)(-3-2i) = i(-6 - 4i - 9i - 6i^2) = i(-6 - 13i + 6) = i(-13i) = -13i^2 = 13$, which is also a real number. Fantastic!

• Scenario 3: Let's pick d = 0.
  If $d = 0$, then using $2c = 3d$, we get $2c = 3(0)$, which implies $c = 0$. So, when both $c$ and $d$ are 0, the expression becomes $i(2+3i)(0+0i) = i(2+3i)(0) = 0$, which is a real number. This is a special case where both variables are zero.

These examples demonstrate that as long as the condition $2c = 3d$ is met, the expression $i(2+3i)(c+di)$ will result in a real number. These scenarios help illustrate the relationship between $c$ and $d$ and provide concrete instances where the condition is satisfied.

The General Solution

We've danced around it, we've given examples, but now let's state the general solution explicitly. We found that the condition for $i(2+3i)(c+di)$ to be a real number is $2c - 3d = 0$. This can be rewritten as $2c = 3d$.

To express the general solution, we can say that for any real number $d$, we can find a corresponding $c$ that satisfies the condition. Let's express $c$ in terms of $d$: c = rac{3}{2}d. Alternatively, we can express $d$ in terms of $c$: d = rac{2}{3}c.

So, the general solution can be represented as pairs $(c, d)$ where c = rac{3}{2}d or d = rac{2}{3}c. This means that there are infinitely many pairs of $c$ and $d$ that make the expression a real number, as long as they satisfy this relationship.

To put it in a more mathematical form, we can say that the set of solutions is:

$\{(c, d) \in \mathbb{R}^2 \mid 2c = 3d\}$

This notation means "the set of all pairs $(c, d)$ where $c$ and $d$ are real numbers, such that $2c = 3d$." This concisely captures all possible pairs of real numbers $c$ and $d$ that make the given expression a real number. This general solution is a powerful statement because it encompasses all specific cases and scenarios we discussed earlier. It also highlights the infinite nature of the solution set, meaning there are countless combinations of $c$ and $d$ that will work.

Conclusion

Alright, guys, we did it! We successfully navigated the world of complex numbers and figured out the values of $c$ and $d$ that make the expression $i(2+3i)(c+di)$ a real number. The key takeaway is that $c$ and $d$ must satisfy the condition $2c = 3d$. This means that $c$ must be rac{3}{2} times $d$, or equivalently, $d$ must be rac{2}{3} times $c$. There isn't just one right answer; instead, there are infinitely many pairs of $c$ and $d$ that work, as long as they adhere to this relationship. We even checked some specific examples to see this in action. This problem showcases the beauty of complex numbers and how seemingly abstract conditions can lead to concrete relationships between variables. So, the next time you see a complex number problem, remember our adventure here, and you'll be well-equipped to tackle it!