What Is The Solution To $x^4 - 12x^2 + 10 \ \textgreater \ 0$?A. $(-0.95, 0.95$\]B. $(-3.33, 3.33$\]C. $(-3.33, -0.95) \cup (0.95, 3.33$\]D. $(-\square, -3.33) \cup (-0.95, 0.95) \cup (3.33, \square$\]

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Introduction

In this article, we will explore the solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$. This inequality involves a quartic polynomial, which can be challenging to solve. We will use various mathematical techniques, including factoring and the quadratic formula, to find the solution to this inequality.

Understanding the Inequality

The given inequality is $x^4 - 12x^2 + 10 \ \textgreater \ 0$. This is a quartic inequality, which means it involves a polynomial of degree 4. The polynomial can be factored as $(x^2 - 6)^2 - 4$. This can be further simplified to $(x^2 - 6)^2 - 2^2$, which is a difference of squares.

Factoring the Polynomial

We can factor the polynomial as $(x^2 - 6)^2 - 2^2 = (x^2 - 6 - 2)(x^2 - 6 + 2) = (x^2 - 8)(x^2 - 4)$. This gives us two quadratic factors, which can be further factored as $(x - \sqrt{8})(x + \sqrt{8})(x - \sqrt{4})(x + \sqrt{4})$.

Solving the Inequality

To solve the inequality, we need to find the values of $x$ that make the polynomial positive. We can do this by finding the values of $x$ that make each of the quadratic factors positive.

Finding the Critical Points

The critical points of the inequality are the values of $x$ that make each of the quadratic factors equal to zero. These points are $x = \pm \sqrt{8}$ and $x = \pm \sqrt{4}$.

Testing the Intervals

We can test the intervals between the critical points by plugging in a test value from each interval into the polynomial. If the polynomial is positive at the test value, then the entire interval is part of the solution.

Finding the Solution

After testing the intervals, we find that the solution to the inequality is $(-\infty, -\sqrt{8}) \cup (-\sqrt{4}, -\sqrt{8}) \cup (\sqrt{8}, \sqrt{4}) \cup (\infty, \sqrt{8})$. However, we can simplify this solution by combining the intervals that are part of the solution.

Simplifying the Solution

We can simplify the solution by combining the intervals that are part of the solution. This gives us the solution $(-\infty, -\sqrt{8}) \cup (-\sqrt{4}, -\sqrt{8}) \cup (\sqrt{8}, \sqrt{4}) \cup (\infty, \sqrt{8}) = (-\infty, -3.33) \cup (-0.95, 0.95) \cup (3.33, \infty)$.

Conclusion

In this article, we have explored the solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$. We have used various mathematical techniques, including factoring and the quadratic formula, to find the solution to this inequality. The solution is $(-\infty, -3.33) \cup (-0.95, 0.95) \cup (3.33, \infty)$.

Final Answer

The final answer to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is $(-\infty, -3.33) \cup (-0.95, 0.95) \cup (3.33, \infty)$. This is option D.

Discussion

The solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is $(-\infty, -3.33) \cup (-0.95, 0.95) \cup (3.33, \infty)$. This is option D. The other options are not correct.

Key Takeaways

• The solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is $(-\infty, -3.33) \cup (-0.95, 0.95) \cup (3.33, \infty)$.
• The solution can be found by factoring the polynomial and using the quadratic formula.
• The critical points of the inequality are $x = \pm \sqrt{8}$ and $x = \pm \sqrt{4}$.
• The solution can be tested by plugging in a test value from each interval into the polynomial.

References

• [1] "Quartic Inequalities" by Math Open Reference
• [2] "Quadratic Formula" by Math Is Fun
• [3] "Factoring Polynomials" by Purplemath

Related Articles

• "Solving Quadratic Inequalities"
• "Factoring Quadratic Expressions"
• "Quartic Polynomials"

Tags

• Inequality
• Quartic Polynomial
• Factoring
• Quadratic Formula
• Critical Points
• Solution
• Math
• Mathematics
  

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Introduction

In our previous article, we explored the solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$. We used various mathematical techniques, including factoring and the quadratic formula, to find the solution to this inequality. In this article, we will answer some frequently asked questions about solving this inequality.

Q: What is the first step in solving the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$?

A: The first step in solving the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is to factor the polynomial. We can factor the polynomial as $(x^2 - 6)^2 - 2^2$, which is a difference of squares.

Q: How do I factor the polynomial $(x^2 - 6)^2 - 2^2$?

A: To factor the polynomial $(x^2 - 6)^2 - 2^2$, we can use the difference of squares formula: $a^2 - b^2 = (a - b)(a + b)$. In this case, $a = x^2 - 6$ and $b = 2$. So, we can factor the polynomial as $(x^2 - 6 - 2)(x^2 - 6 + 2)$.

Q: What are the critical points of the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$?

A: The critical points of the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ are the values of $x$ that make each of the quadratic factors equal to zero. These points are $x = \pm \sqrt{8}$ and $x = \pm \sqrt{4}$.

Q: How do I test the intervals between the critical points?

A: To test the intervals between the critical points, we can plug in a test value from each interval into the polynomial. If the polynomial is positive at the test value, then the entire interval is part of the solution.

Q: What is the solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$?

A: The solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is $(-\infty, -3.33) \cup (-0.95, 0.95) \cup (3.33, \infty)$.

Q: Why is the solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ not $(-\infty, -3.33) \cup (-0.95, 3.33)$?

A: The solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is not $(-\infty, -3.33) \cup (-0.95, 3.33)$ because the interval $(-0.95, 3.33)$ is not part of the solution. This is because the polynomial is negative at the test value $x = 1$, which is in the interval $(-0.95, 3.33)$.

Q: How do I know if the solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is correct?

A: To know if the solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is correct, we can plug in a test value from each interval into the polynomial. If the polynomial is positive at the test value, then the entire interval is part of the solution.

Q: What are some common mistakes to avoid when solving the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$?

A: Some common mistakes to avoid when solving the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ are:

• Not factoring the polynomial correctly
• Not finding the critical points correctly
• Not testing the intervals correctly
• Not checking the solution correctly

Conclusion

In this article, we have answered some frequently asked questions about solving the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$. We have covered topics such as factoring the polynomial, finding the critical points, testing the intervals, and checking the solution. We hope that this article has been helpful in understanding how to solve this inequality.

Final Answer

The final answer to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is $(-\infty, -3.33) \cup (-0.95, 0.95) \cup (3.33, \infty)$.

Discussion

The solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is $(-\infty, -3.33) \cup (-0.95, 0.95) \cup (3.33, \infty)$. This is the correct solution to the inequality.

Key Takeaways

• The first step in solving the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is to factor the polynomial.
• The critical points of the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ are $x = \pm \sqrt{8}$ and $x = \pm \sqrt{4}$.
• The solution to the inequality $x^4 - 12x^2 + 10 \ \textgreater \ 0$ is $(-\infty, -3.33) \cup (-0.95, 0.95) \cup (3.33, \infty)$.

References

• [1] "Quartic Inequalities" by Math Open Reference
• [2] "Quadratic Formula" by Math Is Fun
• [3] "Factoring Polynomials" by Purplemath

Related Articles

• "Solving Quadratic Inequalities"
• "Factoring Quadratic Expressions"
• "Quartic Polynomials"

Tags

• Inequality
• Quartic Polynomial
• Factoring
• Quadratic Formula
• Critical Points
• Solution
• Math
• Mathematics