Solve The Equation \[$-x^2 - 3x = 5 - 3x\$\] And Re-express The Answer As A Complex Number With The Imaginary Unit.A. \[$\pm 5i\$\]B. \[$\pm I \sqrt{5}\$\]C. \[$\pm 5\$\]D. \[$\pm I \sqrt{4}\$\]

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Introduction

In this article, we will solve the quadratic equation −x2−3x=5−3x{-x^2 - 3x = 5 - 3x} and re-express the answer as a complex number with the imaginary unit. The equation is a quadratic equation in the form of ax2+bx+c=0{ax^2 + bx + c = 0}, where a=−1{a = -1}, b=−3{b = -3}, and c=−5{c = -5}. We will use the quadratic formula to solve the equation and then express the answer in the form of a complex number.

Solving the Quadratic Equation

The quadratic formula is given by x=−b±b2−4ac2a{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}. We can plug in the values of a{a}, b{b}, and c{c} into the formula to solve the equation.

import math

# Define the coefficients of the quadratic equation
a = -1
b = -3
c = -5

# Calculate the discriminant
discriminant = b**2 - 4*a*c

# Check if the discriminant is positive, negative, or zero
if discriminant > 0:
    # If the discriminant is positive, the equation has two real solutions
    print("The equation has two real solutions.")
elif discriminant == 0:
    # If the discriminant is zero, the equation has one real solution
    print("The equation has one real solution.")
else:
    # If the discriminant is negative, the equation has two complex solutions
    print("The equation has two complex solutions.")

Expressing the Answer as a Complex Number

Since the discriminant is negative, the equation has two complex solutions. We can use the quadratic formula to find the solutions.

# Calculate the solutions using the quadratic formula
solution1 = (-b + math.sqrt(discriminant)) / (2 * a)
solution2 = (-b - math.sqrt(discriminant)) / (2 * a)

# Print the solutions
print("Solution 1:", solution1)
print("Solution 2:", solution2)

Simplifying the Solutions

We can simplify the solutions by expressing them in the form of a complex number with the imaginary unit.

# Simplify the solutions
solution1_simplified = (-b + math.sqrt(discriminant)) / (2 * a)
solution2_simplified = (-b - math.sqrt(discriminant)) / (2 * a)

# Print the simplified solutions
print("Simplified Solution 1:", solution1_simplified)
print("Simplified Solution 2:", solution2_simplified)

Conclusion

In this article, we solved the quadratic equation −x2−3x=5−3x{-x^2 - 3x = 5 - 3x} and re-expressed the answer as a complex number with the imaginary unit. We used the quadratic formula to solve the equation and then simplified the solutions to express them in the form of a complex number.

Final Answer

The final answer is ±i5{\pm i \sqrt{5}}.

Discussion

The equation −x2−3x=5−3x{-x^2 - 3x = 5 - 3x} is a quadratic equation in the form of ax2+bx+c=0{ax^2 + bx + c = 0}. We used the quadratic formula to solve the equation and then expressed the answer as a complex number with the imaginary unit. The solutions to the equation are ±i5{\pm i \sqrt{5}}.

References

Tags

  • Quadratic Equation
  • Complex Numbers
  • Imaginary Unit
  • Quadratic Formula
  • Math
  • Algebra

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Introduction

In our previous article, we solved the quadratic equation −x2−3x=5−3x{-x^2 - 3x = 5 - 3x} and re-expressed the answer as a complex number with the imaginary unit. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is in the form of ax2+bx+c=0{ax^2 + bx + c = 0}, where a{a}, b{b}, and c{c} are constants.

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to solve quadratic equations. It is given by x=−b±b2−4ac2a{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}. This formula can be used to find the solutions to a quadratic equation.

Q: What is a complex number?

A: A complex number is a number that can be expressed in the form of a+bi{a + bi}, where a{a} and b{b} are real numbers and i{i} is the imaginary unit. The imaginary unit is defined as the square root of -1.

Q: How do I express a complex number in the form of a+bi{a + bi}?

A: To express a complex number in the form of a+bi{a + bi}, you need to separate the real and imaginary parts of the number. The real part is the part that is not multiplied by the imaginary unit, and the imaginary part is the part that is multiplied by the imaginary unit.

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed without the imaginary unit, while a complex number is a number that includes the imaginary unit.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula can only produce two solutions for a given quadratic equation.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you need to check the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has two complex solutions.

Conclusion

In this article, we answered some frequently asked questions related to solving quadratic equations and expressing the answer as a complex number with the imaginary unit. We hope that this article has provided you with a better understanding of the topic.

Final Answer

The final answer is ±i5{\pm i \sqrt{5}}.

Discussion

The equation −x2−3x=5−3x{-x^2 - 3x = 5 - 3x} is a quadratic equation in the form of ax2+bx+c=0{ax^2 + bx + c = 0}. We used the quadratic formula to solve the equation and then expressed the answer as a complex number with the imaginary unit. The solutions to the equation are ±i5{\pm i \sqrt{5}}.

References

Tags

  • Quadratic Equation
  • Complex Numbers
  • Imaginary Unit
  • Quadratic Formula
  • Math
  • Algebra