Solve 3 X 2 + 6 X + 15 = 0 3x^2 + 6x + 15 = 0 3 X 2 + 6 X + 15 = 0 .A. − 1 ± I -1 \pm I − 1 ± I B. − 1 ± 2 I -1 \pm 2i − 1 ± 2 I C. − 2 ± I -2 \pm I − 2 ± I D. − 2 ± 2 I -2 \pm 2i − 2 ± 2 I

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods used to solve them. In this article, we will focus on solving the quadratic equation $3x^2 + 6x + 15 = 0$. We will use the quadratic formula to find the solutions to this equation.

The Quadratic Formula

The quadratic formula is a powerful tool used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

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

In this equation, $a$, $b$, and $c$ are the coefficients of the quadratic equation. To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the given equation.

Identifying the Coefficients

In the equation $3x^2 + 6x + 15 = 0$, we can identify the coefficients as follows:

$a = 3$ $b = 6$ $c = 15$

Applying the Quadratic Formula

Now that we have identified the coefficients, we can apply the quadratic formula to find the solutions to the equation. Plugging in the values of $a$, $b$, and $c$ into the quadratic formula, we get:

$x = \frac{-6 \pm \sqrt{6^2 - 4(3)(15)}}{2(3)}$

Simplifying the Expression

To simplify the expression, we need to evaluate the expression inside the square root. This involves calculating the value of $6^2 - 4(3)(15)$.

$6^2 - 4(3)(15) = 36 - 180 = -144$

Evaluating the Square Root

Now that we have evaluated the expression inside the square root, we can simplify the expression further. The square root of $-144$ is $12i$, where $i$ is the imaginary unit.

$\sqrt{-144} = 12i$

Simplifying the Quadratic Formula

Now that we have evaluated the square root, we can simplify the quadratic formula further. Plugging in the value of the square root into the quadratic formula, we get:

$x = \frac{-6 \pm 12i}{6}$

Simplifying the Expression Further

To simplify the expression further, we can divide both the real and imaginary parts of the numerator by the denominator.

$x = \frac{-6}{6} \pm \frac{12i}{6}$

Evaluating the Expression

Evaluating the expression, we get:

$x = -1 \pm 2i$

Conclusion

In this article, we have solved the quadratic equation $3x^2 + 6x + 15 = 0$ using the quadratic formula. We have identified the coefficients of the equation, applied the quadratic formula, simplified the expression, and evaluated the square root. The solutions to the equation are $-1 \pm 2i$.

Discussion

The solutions to the equation $3x^2 + 6x + 15 = 0$ are complex numbers. Complex numbers are numbers that have both real and imaginary parts. In this case, the solutions are $-1 \pm 2i$, where $i$ is the imaginary unit.

Comparison with Other Options

Let's compare the solutions we obtained with the other options given in the problem.

A. $-1 \pm i$ B. $-1 \pm 2i$ C. $-2 \pm i$ D. $-2 \pm 2i$

Our solutions match option B, which is $-1 \pm 2i$.

Final Answer

The final answer to the problem is $-1 \pm 2i$.

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Introduction

In our previous article, we solved the quadratic equation $3x^2 + 6x + 15 = 0$ using the quadratic formula. We obtained the solutions $-1 \pm 2i$. In this article, we will answer some frequently asked questions related to the solution of this equation.

Q&A

Q1: What is the quadratic formula?

A1: The quadratic formula is a powerful tool used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by:

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

Q2: How do I identify the coefficients of a quadratic equation?

A2: To identify the coefficients of a quadratic equation, you need to look at the equation and identify the values of $a$, $b$, and $c$. In the equation $3x^2 + 6x + 15 = 0$, we can identify the coefficients as follows:

$a = 3$ $b = 6$ $c = 15$

Q3: What is the difference between real and imaginary numbers?

A3: Real numbers are numbers that have no imaginary part. Imaginary numbers are numbers that have an imaginary part. In the solutions $-1 \pm 2i$, the $-1$ is a real number and the $2i$ is an imaginary number.

Q4: Why do we use the quadratic formula to solve quadratic equations?

A4: The quadratic formula is a powerful tool used to solve quadratic equations because it allows us to find the solutions to any quadratic equation, regardless of whether the solutions are real or complex.

Q5: Can you explain the concept of complex numbers?

A5: Complex numbers are numbers that have both real and imaginary parts. They are written in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part. In the solutions $-1 \pm 2i$, the $-1$ is the real part and the $2i$ is the imaginary part.

Q6: How do I simplify complex numbers?

A6: To simplify complex numbers, you need to combine the real and imaginary parts. For example, in the solutions $-1 \pm 2i$, we can combine the real and imaginary parts to get $-1 \pm 2i$.

Q7: Can you explain the concept of the imaginary unit?

A7: The imaginary unit is a mathematical concept that is used to represent the square root of $-1$. It is denoted by the letter $i$. In the solutions $-1 \pm 2i$, the $i$ is the imaginary unit.

Q8: How do I evaluate the square root of a negative number?

A8: To evaluate the square root of a negative number, you need to use the imaginary unit. For example, the square root of $-144$ is $12i$, where $i$ is the imaginary unit.

Q9: Can you explain the concept of quadratic equations with complex solutions?

A9: Quadratic equations with complex solutions are equations that have solutions that are complex numbers. In the equation $3x^2 + 6x + 15 = 0$, we obtained the solutions $-1 \pm 2i$, which are complex numbers.

Q10: How do I determine whether a quadratic equation has real or complex solutions?

A10: To determine whether a quadratic equation has real or complex solutions, you need to look at the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has real solutions. If the discriminant is negative, the equation has complex solutions.

Conclusion

In this article, we have answered some frequently asked questions related to the solution of the quadratic equation $3x^2 + 6x + 15 = 0$. We have explained the concept of the quadratic formula, identified the coefficients of the equation, and simplified complex numbers. We have also explained the concept of complex numbers, the imaginary unit, and quadratic equations with complex solutions.