Solve The Equation $x^4 - 17x^2 + 16 = 0$.Let $u = X^2$. Rewrite The Equation In Terms Of $u$: $u^2 - 17u + 16 = 0$.Factor The Equation:$(u - 16)(u - 1) = 0$.Select All Of The Solutions To The Original

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Introduction

In this article, we will solve the equation $x^4 - 17x^2 + 16 = 0$ using a substitution method. We will let $u = x^2$ and rewrite the equation in terms of $u$. Then, we will factor the equation and find the solutions to the original equation.

Step 1: Substitute $u = x^2$

Let $u = x^2$. This means that $x^4 = (x^2)^2 = u^2$ and $x^2 = u$. Substituting these expressions into the original equation, we get:

$$u^2 - 17u + 16 = 0$$

This is a quadratic equation in terms of $u$.

Step 2: Factor the Equation

The equation $u^2 - 17u + 16 = 0$ can be factored as:

$$(u - 16)(u - 1) = 0$$

This means that either $(u - 16) = 0$ or $(u - 1) = 0$.

Step 3: Solve for $u$

Solving for $u$, we get:

$$u - 16 = 0 \Rightarrow u = 16$$

$$u - 1 = 0 \Rightarrow u = 1$$

Step 4: Substitute Back $u = x^2$

Now, we substitute back $u = x^2$ to get:

$$x^2 = 16 \Rightarrow x = \pm 4$$

$$x^2 = 1 \Rightarrow x = \pm 1$$

Conclusion

In this article, we solved the equation $x^4 - 17x^2 + 16 = 0$ using a substitution method. We let $u = x^2$ and rewrote the equation in terms of $u$. Then, we factored the equation and found the solutions to the original equation. The solutions to the original equation are $x = \pm 4$ and $x = \pm 1$.

Discussion

The equation $x^4 - 17x^2 + 16 = 0$ is a quadratic equation in terms of $x^2$. We can solve it by letting $u = x^2$ and rewriting the equation in terms of $u$. Then, we can factor the equation and find the solutions to the original equation.

Example

Suppose we want to solve the equation $x^4 - 18x^2 + 25 = 0$. We can let $u = x^2$ and rewrite the equation in terms of $u$:

$$u^2 - 18u + 25 = 0$$

This equation can be factored as:

$$(u - 25)(u - 1) = 0$$

Solving for $u$, we get:

$$u - 25 = 0 \Rightarrow u = 25$$

$$u - 1 = 0 \Rightarrow u = 1$$

Substituting back $u = x^2$, we get:

$$x^2 = 25 \Rightarrow x = \pm 5$$

$$x^2 = 1 \Rightarrow x = \pm 1$$

Therefore, the solutions to the equation $x^4 - 18x^2 + 25 = 0$ are $x = \pm 5$ and $x = \pm 1$.

Applications

The equation $x^4 - 17x^2 + 16 = 0$ has many applications in mathematics and science. For example, it can be used to model the motion of a particle in a quadratic potential. It can also be used to solve problems in physics, engineering, and computer science.

Conclusion

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Q: What is the equation $x^4 - 17x^2 + 16 = 0$?

A: The equation $x^4 - 17x^2 + 16 = 0$ is a quadratic equation in terms of $x^2$. It can be rewritten as $u^2 - 17u + 16 = 0$, where $u = x^2$.

Q: How do I solve the equation $x^4 - 17x^2 + 16 = 0$?

A: To solve the equation $x^4 - 17x^2 + 16 = 0$, you can use a substitution method. Let $u = x^2$ and rewrite the equation in terms of $u$. Then, factor the equation and find the solutions to the original equation.

Q: What is the substitution method?

A: The substitution method is a technique used to solve equations by substituting a new variable for an existing variable. In this case, we let $u = x^2$ and rewrite the equation in terms of $u$.

Q: How do I factor the equation $u^2 - 17u + 16 = 0$?

A: The equation $u^2 - 17u + 16 = 0$ can be factored as $(u - 16)(u - 1) = 0$. This means that either $(u - 16) = 0$ or $(u - 1) = 0$.

Q: What are the solutions to the equation $x^4 - 17x^2 + 16 = 0$?

A: The solutions to the equation $x^4 - 17x^2 + 16 = 0$ are $x = \pm 4$ and $x = \pm 1$.

Q: Can I use the substitution method to solve other equations?

A: Yes, you can use the substitution method to solve other equations. For example, you can use it to solve the equation $x^4 - 18x^2 + 25 = 0$.

Q: What are some applications of the equation $x^4 - 17x^2 + 16 = 0$?

A: The equation $x^4 - 17x^2 + 16 = 0$ has many applications in mathematics and science. For example, it can be used to model the motion of a particle in a quadratic potential. It can also be used to solve problems in physics, engineering, and computer science.

Q: How do I know when to use the substitution method?

A: You can use the substitution method when you have an equation that can be rewritten in terms of a new variable. In this case, we let $u = x^2$ and rewrote the equation in terms of $u$.

Q: What are some common mistakes to avoid when using the substitution method?

A: Some common mistakes to avoid when using the substitution method include:

• Not checking if the equation can be rewritten in terms of a new variable
• Not factoring the equation correctly
• Not substituting back the original variable correctly

Q: Can I use the substitution method to solve equations with higher degrees?

A: Yes, you can use the substitution method to solve equations with higher degrees. However, you may need to use more advanced techniques, such as the quadratic formula or the rational root theorem.

Conclusion

In this article, we answered some common questions about solving the equation $x^4 - 17x^2 + 16 = 0$ using the substitution method. We discussed the substitution method, factoring the equation, and finding the solutions to the original equation. We also provided some examples and applications of the equation.