Find The Solutions Of $x^4+5x^2-36=0$. Select All That Apply.A) $-3i$ B) $-2i$ C) $2i$ D) $3i$ E) -3 F) -2 G) 2 H) 3

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Introduction

Quartic equations are a type of polynomial equation of degree four, which can be challenging to solve. In this article, we will focus on solving the quartic equation $x^4+5x^2-36=0$. We will explore various methods to find the solutions of this equation and identify the correct options among the given choices.

Understanding the Equation

The given equation is a quartic equation in the form of $ax^4+bx^2+c=0$, where $a=1$, $b=5$, and $c=-36$. To solve this equation, we can use various methods such as factoring, substitution, or numerical methods.

Factoring the Equation

One way to solve the equation is to factor it. We can start by noticing that the equation has a common factor of $x^2$. We can rewrite the equation as $(x^2)^2+5(x^2)-36=0$. This can be further simplified to $(x^2+9)(x^2-4)=0$.

Solving the Quadratic Factors

Now, we have two quadratic factors: $x^2+9=0$ and $x^2-4=0$. We can solve these factors separately.

Solving $x^2+9=0$

To solve the equation $x^2+9=0$, we can use the quadratic formula: $x=\pm\sqrt{-9}$. Since the square root of a negative number is an imaginary number, we have $x=\pm3i$.

Solving $x^2-4=0$

To solve the equation $x^2-4=0$, we can use the quadratic formula: $x=\pm\sqrt{4}$. This gives us $x=\pm2$.

Combining the Solutions

Now, we have the solutions of the two quadratic factors. We can combine these solutions to find the solutions of the original equation.

Analyzing the Options

We are given several options to choose from. Let's analyze each option and determine if it is a solution of the equation.

Option A: $-3i$

We have already found that $x=\pm3i$ is a solution of the equation. Therefore, option A is a correct solution.

Option B: $-2i$

We have not found any solution of the form $-2i$. Therefore, option B is not a correct solution.

Option C: $2i$

We have already found that $x=\pm3i$ is a solution of the equation, but not $2i$. Therefore, option C is not a correct solution.

Option D: $3i$

We have already found that $x=\pm3i$ is a solution of the equation. Therefore, option D is a correct solution.

Option E: $-3$

We have found that $x=\pm2$ is a solution of the equation, but not $-3$. Therefore, option E is not a correct solution.

Option F: $-2$

We have found that $x=\pm2$ is a solution of the equation. Therefore, option F is a correct solution.

Option G: $2$

We have found that $x=\pm2$ is a solution of the equation. Therefore, option G is a correct solution.

Option H: $3$

We have not found any solution of the form $3$. Therefore, option H is not a correct solution.

Conclusion

In conclusion, the correct solutions of the equation $x^4+5x^2-36=0$ are $-3i$, $3i$, $-2$, and $2$. These solutions can be obtained by factoring the equation and solving the resulting quadratic factors.

Final Answer

The final answer is:

• A) $-3i$
• D) $3i$
• F) $-2$
• G) $2$

Note: The options are selected based on the solutions obtained in the previous sections.

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Introduction

In our previous article, we solved the quartic equation $x^4+5x^2-36=0$ and found the solutions to be $-3i$, $3i$, $-2$, and $2$. In this article, we will answer some frequently asked questions related to this equation.

Q&A

Q: What is a quartic equation?

A: A quartic equation is a type of polynomial equation of degree four, which can be written in the form of $ax^4+bx^3+cx^2+dx+e=0$. In the case of the equation $x^4+5x^2-36=0$, the coefficient of $x^3$ is zero, making it a special type of quartic equation.

Q: How do I solve a quartic equation?

A: There are several methods to solve a quartic equation, including factoring, substitution, and numerical methods. In the case of the equation $x^4+5x^2-36=0$, we used factoring to solve it.

Q: What is the difference between a quadratic and a quartic equation?

A: A quadratic equation is a type of polynomial equation of degree two, which can be written in the form of $ax^2+bx+c=0$. A quartic equation, on the other hand, is a type of polynomial equation of degree four, which can be written in the form of $ax^4+bx^3+cx^2+dx+e=0$. The main difference between the two is the degree of the equation.

Q: Can I use the quadratic formula to solve a quartic equation?

A: No, the quadratic formula is used to solve quadratic equations, not quartic equations. The quadratic formula is given by $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$, which is only applicable to quadratic equations.

Q: How do I determine if a quartic equation can be factored?

A: To determine if a quartic equation can be factored, you can try to find two binomials whose product is equal to the quartic equation. If you can find such binomials, then the equation can be factored.

Q: What is the significance of the imaginary solutions in the equation $x^4+5x^2-36=0$?

A: The imaginary solutions in the equation $x^4+5x^2-36=0$ are $-3i$ and $3i$. These solutions are important because they indicate that the equation has complex roots.

Q: Can I use numerical methods to solve a quartic equation?

A: Yes, numerical methods can be used to solve a quartic equation. However, these methods may not always provide an exact solution and may require a lot of computational effort.

Conclusion

In conclusion, the quartic equation $x^4+5x^2-36=0$ has solutions of $-3i$, $3i$, $-2$, and $2$. We have also answered some frequently asked questions related to this equation, including how to solve a quartic equation, the difference between a quadratic and a quartic equation, and the significance of the imaginary solutions.

Final Answer

The final answer is:

• A: $-3i$
• D: $3i$
• F: $-2$
• G: $2$

Note: The options are selected based on the solutions obtained in the previous sections.