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

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Introduction

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Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a quadratic equation with complex roots. We will use the given equation $x^2 + 14x + 17 = -96$ and solve for $x$.

Understanding Complex Roots

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Complex roots are a type of root that occurs when the discriminant of a quadratic equation is negative. The discriminant is the expression under the square root in the quadratic formula. When the discriminant is negative, the quadratic formula will produce complex roots.

The Quadratic Formula

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

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Applying the Quadratic Formula

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Let's apply the quadratic formula to the given equation $x^2 + 14x + 17 = -96$. First, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$. We can do this by adding 96 to both sides of the equation:

$x^2 + 14x + 17 + 96 = 0$

This simplifies to:

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

Now we can identify the coefficients $a$, $b$, and $c$. We have $a = 1$, $b = 14$, and $c = 113$.

Calculating the Discriminant

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The discriminant is the expression under the square root in the quadratic formula. It is given by:

$b^2 - 4ac$

In this case, we have:

$14^2 - 4(1)(113)$

This simplifies to:

$196 - 452$

Which is equal to:

$-256$

Solving for $x$

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Now that we have the discriminant, we can solve for $x$ using the quadratic formula:

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

Substituting the values of $a$, $b$, and $c$, we get:

$x = \frac{-14 \pm \sqrt{-256}}{2(1)}$

The square root of $-256$ is a complex number, which can be written as:

$\sqrt{-256} = 16i$

So, we have:

$x = \frac{-14 \pm 16i}{2}$

Simplifying, we get:

$x = -7 \pm 8i$

Conclusion

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In this article, we solved a quadratic equation with complex roots using the quadratic formula. We applied the formula to the given equation and calculated the discriminant. We then solved for $x$ and found that the solutions are complex numbers.

Answer

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The correct answer is:

A. $x = -7 \pm 8i$

This is the only option that matches the solution we found using the quadratic formula.

Discussion

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This problem is a great example of how to solve quadratic equations with complex roots. It requires a good understanding of the quadratic formula and the concept of complex numbers. If you have any questions or need further clarification, please don't hesitate to ask.

Related Topics

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• Quadratic equations with real roots
• Complex numbers
• Quadratic formula
• Discriminant

Practice Problems

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1. Solve the quadratic equation $x^2 + 10x + 24 = 0$.
2. Find the roots of the quadratic equation $x^2 - 6x + 8 = 0$.
3. Solve the quadratic equation $x^2 + 2x - 15 = 0$.

References

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• [1] "Quadratic Equations" by Math Open Reference
• [2] "Complex Numbers" by Khan Academy
• [3] "Quadratic Formula" by Wolfram MathWorld
  

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Introduction

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In our previous article, we solved a quadratic equation with complex roots using the quadratic formula. We also discussed the concept of complex numbers and the discriminant. In this article, we will answer some frequently asked questions about quadratic equations with complex roots.

Q&A

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Q: What is the difference between real and complex roots?

A: Real roots are solutions to a quadratic equation that are real numbers, whereas complex roots are solutions that are complex numbers. Complex roots occur when the discriminant of a quadratic equation is negative.

Q: How do I know if a quadratic equation has complex roots?

A: To determine if a quadratic equation has complex roots, you need to calculate the discriminant. If the discriminant is negative, then the equation has complex roots.

Q: What is the quadratic formula?

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

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

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

A: To apply the quadratic formula, you need to identify the coefficients $a$, $b$, and $c$ of the quadratic equation. Then, you can plug these values into the quadratic formula and solve for $x$.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. It is given by:

$b^2 - 4ac$

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you need to substitute the values of $a$, $b$, and $c$ into the expression $b^2 - 4ac$.

Q: What happens if the discriminant is negative?

A: If the discriminant is negative, then the quadratic equation has complex roots.

Q: How do I solve for $x$ when the discriminant is negative?

A: To solve for $x$ when the discriminant is negative, you need to use the quadratic formula and substitute the values of $a$, $b$, and $c$ into the formula.

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

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

• Not calculating the discriminant correctly
• Not using the correct values of $a$, $b$, and $c$ in the quadratic formula
• Not simplifying the expression under the square root correctly
• Not using the correct sign for the square root (i.e., $\pm$)

Conclusion

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In this article, we answered some frequently asked questions about quadratic equations with complex roots. We discussed the concept of complex numbers, the discriminant, and the quadratic formula. We also provided some tips and common mistakes to avoid when solving quadratic equations with complex roots.

Practice Problems

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1. Solve the quadratic equation $x^2 + 10x + 24 = 0$.
2. Find the roots of the quadratic equation $x^2 - 6x + 8 = 0$.
3. Solve the quadratic equation $x^2 + 2x - 15 = 0$.

References

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• [1] "Quadratic Equations" by Math Open Reference
• [2] "Complex Numbers" by Khan Academy
• [3] "Quadratic Formula" by Wolfram MathWorld

Additional Resources

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• [1] "Quadratic Equations with Complex Roots" by Mathway
• [2] "Complex Numbers and Quadratic Equations" by Purplemath
• [3] "Quadratic Formula and Complex Roots" by IXL