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 used to model a wide range of real-world problems. However, not all quadratic equations have real solutions. In this article, we will explore how to solve quadratic equations that have complex solutions.

What are Complex Solutions?

Complex solutions are solutions to a quadratic equation that involve imaginary numbers. Imaginary numbers are numbers that, when squared, give a negative result. In other words, they are numbers that cannot be expressed on the real number line.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where aa, bb, and cc are the coefficients of the quadratic equation.

Solving the Given Equation

The given equation is:

x2+14x+17=−96x^2 + 14x + 17 = -96

We can rewrite this equation as:

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

Now, we can use the quadratic formula to solve for xx.

Applying the Quadratic Formula

Using the quadratic formula, we get:

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

Simplifying the expression under the square root, we get:

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 seen how to solve a quadratic equation that has complex solutions. We used the quadratic formula to find the solutions, and we obtained two complex solutions: x=−7±8ix = -7 \pm 8i.

Comparison of Solutions

Let's compare our solution with the given options:

  • Option A: x=−7±46ix = -7 \pm 4\sqrt{6}i
  • Option B: x=−7±8ix = -7 \pm 8i
  • Option C: x=7±46ix = 7 \pm 4\sqrt{6}i
  • Option D: x=7±8ix = 7 \pm 8i

Our solution matches with option B: x=−7±8ix = -7 \pm 8i.

Final Answer

The final answer is: B\boxed{B}

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Introduction

In our previous article, we explored how to solve quadratic equations that have complex solutions. We used the quadratic formula to find the solutions and obtained two complex solutions: x=−7±8ix = -7 \pm 8i. In this article, we will answer some frequently asked questions about solving quadratic equations with complex solutions.

Q&A

Q1: What is the difference between real and complex solutions?

A1: Real solutions are solutions to a quadratic equation that can be expressed on the real number line. Complex solutions, on the other hand, involve imaginary numbers and cannot be expressed on the real number line.

Q2: How do I know if a quadratic equation has complex solutions?

A2: To determine if a quadratic equation has complex solutions, you can use the discriminant, which is given by b2−4acb^2 - 4ac. If the discriminant is negative, then the equation has complex solutions.

Q3: What is the quadratic formula?

A3: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where aa, bb, and cc are the coefficients of the quadratic equation.

Q4: How do I apply the quadratic formula to solve a quadratic equation?

A4: To apply the quadratic formula, you need to plug in the values of aa, bb, and cc into the formula. Then, simplify the expression under the square root and solve for xx.

Q5: What if the quadratic equation has complex solutions? How do I handle them?

A5: If the quadratic equation has complex solutions, you can handle them by using the quadratic formula and simplifying the expression under the square root. You may need to use imaginary numbers to express the solutions.

Q6: Can I use the quadratic formula to solve quadratic equations with complex solutions?

A6: Yes, you can use the quadratic formula to solve quadratic equations with complex solutions. The quadratic formula is a powerful tool that can be used to solve quadratic equations, regardless of whether they have real or complex solutions.

Q7: How do I know if my solution is correct?

A7: To verify your solution, you can plug it back into the original equation and check if it satisfies the equation. If it does, then your solution is correct.

Q8: What if I get complex solutions that involve square roots of negative numbers?

A8: If you get complex solutions that involve square roots of negative numbers, you can simplify the expression by using the fact that i2=−1i^2 = -1. This will help you to express the solutions in a more simplified form.

Conclusion

In this article, we have answered some frequently asked questions about solving quadratic equations with complex solutions. We have seen how to use the quadratic formula to find the solutions and how to handle complex solutions. We have also seen how to verify our solutions and simplify complex expressions.

Final Tips

  • Always check the discriminant to determine if the equation has real or complex solutions.
  • Use the quadratic formula to find the solutions.
  • Simplify the expression under the square root and handle complex solutions by using imaginary numbers.
  • Verify your solution by plugging it back into the original equation.

By following these tips, you can become proficient in solving quadratic equations with complex solutions.