Solve: M 4 − 9 M 2 + 8 = 0 M^4 - 9m^2 + 8 = 0 M 4 − 9 M 2 + 8 = 0

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Introduction

Algebraic 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 specific type of algebraic equation, namely a quartic equation, which is a polynomial equation of degree four. The equation we will be solving is $m^4 - 9m^2 + 8 = 0$. This equation is a classic example of a quartic equation, and solving it will require us to employ various algebraic techniques.

Understanding the Equation

The given equation is a quartic equation in the variable $m$. It can be written in the form $ax^4 + bx^3 + cx^2 + dx + e = 0$, where $a = 1$, $b = 0$, $c = -9$, $d = 0$, and $e = 8$. The equation has no linear or quadratic terms, which makes it a special type of quartic equation.

Factoring the Equation

One of the most common methods for solving algebraic equations is factoring. Factoring involves expressing the equation as a product of simpler equations, which can then be solved individually. In this case, we can try to factor the given equation by looking for two binomials whose product is equal to the original equation.

Step 1: Factor the Equation

Let's try to factor the equation by looking for two binomials whose product is equal to the original equation. We can start by looking for two binomials of the form $(m^2 + p)(m^2 + q)$, where $p$ and $q$ are constants.

Step 3: Expand the Product

If we expand the product $(m^2 + p)(m^2 + q)$, we get:

$(m^2 + p)(m^2 + q) = m^4 + (p + q)m^2 + pq$

Step 4: Compare Coefficients

We can compare the coefficients of the expanded product with the original equation to find the values of $p$ and $q$. We have:

$m^4 + (p + q)m^2 + pq = m^4 - 9m^2 + 8$

Step 5: Solve for p and q

Comparing the coefficients of the $m^2$ term, we get:

$p + q = -9$

Comparing the constant term, we get:

$pq = 8$

Step 6: Solve the System of Equations

We now have a system of two equations in two variables:

$p + q = -9$ $pq = 8$

We can solve this system of equations by substitution or elimination. Let's use substitution.

Step 7: Solve for p

From the first equation, we can express $q$ in terms of $p$:

$q = -9 - p$

Step 8: Substitute q into the Second Equation

Substituting $q = -9 - p$ into the second equation, we get:

$p(-9 - p) = 8$

Step 9: Expand and Simplify

Expanding and simplifying the equation, we get:

$-9p - p^2 = 8$

Step 10: Rearrange the Equation

Rearranging the equation, we get:

$p^2 + 9p + 8 = 0$

Step 11: Factor the Quadratic Equation

Factoring the quadratic equation, we get:

$(p + 1)(p + 8) = 0$

Step 12: Solve for p

Solving for $p$, we get:

$p + 1 = 0$ or $p + 8 = 0$

Step 13: Solve for p

Solving for $p$, we get:

$p = -1$ or $p = -8$

Step 14: Find the Corresponding Values of q

Substituting $p = -1$ into the equation $q = -9 - p$, we get:

$q = -9 - (-1) = -8$

Substituting $p = -8$ into the equation $q = -9 - p$, we get:

$q = -9 - (-8) = -1$

Step 15: Write the Factored Form of the Equation

We can now write the factored form of the equation as:

$(m^2 - 1)(m^2 - 8) = 0$

Step 16: Solve the Factored Equation

We can now solve the factored equation by setting each factor equal to zero:

$m^2 - 1 = 0$ or $m^2 - 8 = 0$

Step 17: Solve for m

Solving for $m$, we get:

$m^2 = 1$ or $m^2 = 8$

Step 18: Take the Square Root

Taking the square root of both sides, we get:

$m = \pm 1$ or $m = \pm \sqrt{8}$

Step 19: Simplify the Square Root

Simplifying the square root, we get:

$m = \pm 1$ or $m = \pm 2\sqrt{2}$

Conclusion

In this article, we solved the quartic equation $m^4 - 9m^2 + 8 = 0$ using factoring and the quadratic formula. We first factored the equation into two binomials, and then solved each binomial separately. We found that the solutions to the equation are $m = \pm 1$ and $m = \pm 2\sqrt{2}$. These solutions can be verified by substituting them back into the original equation.

Final Answer

The final answer is $\boxed{m = \pm 1, \pm 2\sqrt{2}}$.

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Introduction

In our previous article, we solved the quartic equation $m^4 - 9m^2 + 8 = 0$ using factoring and the quadratic formula. In this article, we will answer some frequently asked questions about solving this equation.

Q: What is a quartic equation?

A: A quartic equation is a polynomial equation of degree four, which means that the highest power of the variable is four. In this case, the equation is $m^4 - 9m^2 + 8 = 0$.

Q: Why is factoring a good method for solving quartic equations?

A: Factoring is a good method for solving quartic equations because it allows us to break down the equation into simpler equations that can be solved individually. In this case, we were able to factor the equation into two binomials, $(m^2 - 1)$ and $(m^2 - 8)$.

Q: How do I know if an equation can be factored?

A: To determine if an equation can be factored, we need to look for two binomials whose product is equal to the original equation. In this case, we were able to find two binomials, $(m^2 - 1)$ and $(m^2 - 8)$, whose product was equal to the original equation.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations of the form $ax^2 + bx + c = 0$. The quadratic formula is given by:

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

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

A: To use the quadratic formula to solve a quadratic equation, we need to identify the values of $a$, $b$, and $c$ in the equation. We then plug these values into the quadratic formula and simplify.

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

A: No, the quadratic formula is only used to solve quadratic equations, not quartic equations. In this case, we used factoring to solve the quartic equation.

Q: What are the solutions to the quartic equation $m^4 - 9m^2 + 8 = 0$?

A: The solutions to the quartic equation $m^4 - 9m^2 + 8 = 0$ are $m = \pm 1$ and $m = \pm 2\sqrt{2}$.

Q: How do I verify the solutions to a quartic equation?

A: To verify the solutions to a quartic equation, we need to substitute each solution back into the original equation and check if it is true. In this case, we verified that $m = \pm 1$ and $m = \pm 2\sqrt{2}$ are indeed solutions to the equation.

Q: What are some common mistakes to avoid when solving quartic equations?

A: Some common mistakes to avoid when solving quartic equations include:

• Not factoring the equation correctly
• Not using the correct method for solving the equation (e.g. using the quadratic formula for a quartic equation)
• Not verifying the solutions to the equation

Conclusion

In this article, we answered some frequently asked questions about solving the quartic equation $m^4 - 9m^2 + 8 = 0$. We covered topics such as factoring, the quadratic formula, and verifying solutions. We hope that this article has been helpful in answering your questions about solving quartic equations.

Final Answer

The final answer is $\boxed{m = \pm 1, \pm 2\sqrt{2}}$.