Solve For $x$ In The Equation $x^2 + 14x + 17 = -96$.A. $ X = − 7 ± 4 6 I X = -7 \pm 4\sqrt{6}i X = − 7 ± 4 6 ​ I [/tex] B. $x = -7 \pm 8i$ C. $x = 7 \pm 4\sqrt{6}i$ D. $ X = 7 ± 8 I X = 7 \pm 8i X = 7 ± 8 I [/tex]

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they can be solved using various methods. However, when the solutions involve complex numbers, it can be challenging to determine the correct answer. In this article, we will explore how to solve quadratic equations with complex solutions and provide a step-by-step guide to solving the equation $x^2 + 14x + 17 = -96$.

Understanding Complex Numbers

Before we dive into solving the equation, it's essential to understand complex numbers. A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies $i^2 = -1$. Complex numbers can be represented on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Solving the Equation

Now that we have the quadratic formula, let's apply it to the equation $x^2 + 14x + 17 = -96$. First, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$.

x2+14x+17=−96x^2 + 14x + 17 = -96

x2+14x+113=0x^2 + 14x + 113 = 0

Now we can apply the quadratic formula:

x=−14±142−4(1)(113)2(1)x = \frac{-14 \pm \sqrt{14^2 - 4(1)(113)}}{2(1)}

x=−14±196−4522x = \frac{-14 \pm \sqrt{196 - 452}}{2}

x=−14±−2562x = \frac{-14 \pm \sqrt{-256}}{2}

x=−14±16i2x = \frac{-14 \pm 16i}{2}

x=−7±8ix = -7 \pm 8i

Conclusion

In this article, we have solved the quadratic equation $x^2 + 14x + 17 = -96$ using the quadratic formula. The solutions involve complex numbers, and we have shown that the correct answer is $x = -7 \pm 8i$. This example demonstrates the importance of understanding complex numbers and the quadratic formula in solving quadratic equations.

Comparison of Options

Now that we have solved the equation, let's compare our answer with the options provided:

  • A. $x = -7 \pm 4\sqrt{6}i$
  • B. $x = -7 \pm 8i$
  • C. $x = 7 \pm 4\sqrt{6}i$
  • D. $x = 7 \pm 8i$

Our answer, $x = -7 \pm 8i$, matches option B. Therefore, the correct answer is:

The Correct Answer is B. $x = -7 \pm 8i$

Final Thoughts

Introduction

In our previous article, we explored how to solve quadratic equations with complex solutions. We applied the quadratic formula to the equation $x^2 + 14x + 17 = -96$ and obtained the solutions $x = -7 \pm 8i$. In this article, we will provide a Q&A guide to help you better understand quadratic equations with complex solutions.

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, which satisfies $i^2 = -1$.

Q: How do I represent complex numbers on a complex plane?

A: Complex numbers can be represented on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do I apply the quadratic formula to a quadratic equation?

A: To apply the quadratic formula, you need to rewrite the equation in the standard form $ax^2 + bx + c = 0$. Then, you can plug the values of $a$, $b$, and $c$ into the quadratic formula and simplify.

Q: What happens when the discriminant is negative?

A: When the discriminant is negative, the solutions will involve complex numbers. In this case, you will need to use the quadratic formula to obtain the solutions.

Q: How do I simplify complex solutions?

A: To simplify complex solutions, you can use the following steps:

  1. Multiply the numerator and denominator by the conjugate of the denominator.
  2. Simplify the expression.
  3. Write the solution in the form $a + bi$.

Q: What are some common mistakes to avoid when solving quadratic equations with complex solutions?

A: Some common mistakes to avoid when solving quadratic equations with complex solutions include:

  1. Not rewriting the equation in the standard form $ax^2 + bx + c = 0$.
  2. Not using the quadratic formula correctly.
  3. Not simplifying the complex solutions correctly.

Q: How do I check my solutions?

A: To check your solutions, you can plug the values back into the original equation and simplify. If the equation is true, then the solution is correct.

Conclusion

In this article, we have provided a Q&A guide to help you better understand quadratic equations with complex solutions. By following the steps outlined in this article, you can solve quadratic equations with complex solutions and gain a deeper understanding of this fundamental concept in mathematics.

Additional Resources

If you need additional help or resources, here are some suggestions:

  • Khan Academy: Quadratic Equations with Complex Solutions
  • Mathway: Quadratic Equations with Complex Solutions
  • Wolfram Alpha: Quadratic Equations with Complex Solutions

Final Thoughts

Solving quadratic equations with complex solutions requires a deep understanding of complex numbers and the quadratic formula. By following the steps outlined in this article and using the resources provided, you can solve quadratic equations with complex solutions and gain a deeper understanding of this fundamental concept in mathematics.