Factor Completely: $x^4 - 2$2. Solve: $x^2 = 14x - 45$3. The Height Of A Triangle Is [Complete The Sentence Or Provide Additional Information For This Statement To Make Sense.]

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Introduction

In this article, we will delve into two mathematical problems: factoring the expression $x^4 - 2$ and solving the quadratic equation $x^2 = 14x - 45$. These problems require a combination of algebraic techniques and mathematical reasoning to arrive at the solutions. We will break down each problem step by step, providing explanations and examples to help readers understand the concepts.

Factor Completely: $x^4 - 2$

Understanding the Problem

The problem asks us to factor the expression $x^4 - 2$. Factoring an expression means expressing it as a product of simpler expressions, called factors. In this case, we need to find two binomial factors that, when multiplied together, give us the original expression.

Step 1: Identify the Difference of Squares

The expression $x^4 - 2$ can be rewritten as $(x^2)^2 - 2$. This is a difference of squares, which is a special case of factoring. A difference of squares can be factored into the product of two binomials: $(a^2 - b^2) = (a - b)(a + b)$.

Step 2: Apply the Difference of Squares Formula

Using the difference of squares formula, we can rewrite the expression as $(x^2 - \sqrt{2})(x^2 + \sqrt{2})$. However, this is not the final answer, as we need to factor the expression completely.

Step 3: Factor the Quadratic Expressions

The expression $x^2 - \sqrt{2}$ can be factored as $(x - \sqrt[4]{2})(x + \sqrt[4]{2})$. Similarly, the expression $x^2 + \sqrt{2}$ can be factored as $(x - i\sqrt[4]{2})(x + i\sqrt[4]{2})$, where $i$ is the imaginary unit.

Step 4: Write the Final Answer

Combining the results from the previous steps, we can write the final answer as:

$$x^4 - 2 = (x - \sqrt[4]{2})(x + \sqrt[4]{2})(x - i\sqrt[4]{2})(x + i\sqrt[4]{2})$$

Solve: $x^2 = 14x - 45$

Understanding the Problem

The problem asks us to solve the quadratic equation $x^2 = 14x - 45$. A quadratic equation is a polynomial equation of degree two, which means it has a highest power of two. In this case, we need to find the values of $x$ that satisfy the equation.

Step 1: Rearrange the Equation

To solve the equation, we need to rearrange it to the standard form $ax^2 + bx + c = 0$. We can do this by subtracting $14x$ from both sides and adding $45$ to both sides:

$$x^2 - 14x + 45 = 0$$

Step 2: Factor the Quadratic Expression

The quadratic expression $x^2 - 14x + 45$ can be factored as $(x - 9)(x - 5)$. This is a product of two binomials, which means we can set each binomial equal to zero to find the solutions.

Step 3: Solve for $x$

Setting each binomial equal to zero, we get:

$$x - 9 = 0 \Rightarrow x = 9$$

$$x - 5 = 0 \Rightarrow x = 5$$

Step 4: Write the Final Answer

Combining the results from the previous steps, we can write the final answer as:

$$x = 9 \text{ or } x = 5$$

The Height of a Triangle

Understanding the Problem

The problem asks us to find the height of a triangle. However, the statement is incomplete, and we need more information to provide a solution.

Discussion

To find the height of a triangle, we need to know the base and the area of the triangle. The area of a triangle can be calculated using the formula $A = \frac{1}{2}bh$, where $A$ is the area, $b$ is the base, and $h$ is the height. We can rearrange this formula to solve for $h$:

$$h = \frac{2A}{b}$$

However, without more information about the triangle, we cannot provide a specific solution.

Conclusion

In this article, we have solved two mathematical problems: factoring the expression $x^4 - 2$ and solving the quadratic equation $x^2 = 14x - 45$. We have also discussed the problem of finding the height of a triangle, but we need more information to provide a solution. We hope that this article has provided a helpful guide to these mathematical concepts.

Introduction

In our previous article, we delved into the world of mathematics, exploring the concepts of factoring and quadratic equations. We also touched on the topic of finding the height of a triangle, but left it as an open question. In this article, we will continue to explore mathematical concepts, answering some of the most frequently asked questions in the field.

Q&A: Factoring and Quadratic Equations

Q: What is factoring, and how do I do it?

A: Factoring is the process of expressing an algebraic expression as a product of simpler expressions, called factors. To factor an expression, you need to identify the greatest common factor (GCF) of the terms and then use the distributive property to rewrite the expression as a product of factors.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two binomials whose product is equal to the original expression. You can use the quadratic formula to find the roots of the expression, and then use those roots to write the expression as a product of binomials.

Q: What is the quadratic formula, and how do I use it?

A: The quadratic formula is a mathematical formula that is used to find the roots of a quadratic equation. The formula is:

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

To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ from the quadratic equation, and then simplify the expression to find the roots.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to find the values of $x$ that satisfy the equation. You can use the quadratic formula to find the roots of the equation, or you can factor the equation and set each factor equal to zero to find the solutions.

Q&A: Geometry and Trigonometry

Q: What is the formula for the area of a triangle?

A: The formula for the area of a triangle is:

$$A = \frac{1}{2}bh$$

Where $A$ is the area, $b$ is the base, and $h$ is the height.

Q: How do I find the height of a triangle?

A: To find the height of a triangle, you need to know the base and the area of the triangle. You can use the formula $h = \frac{2A}{b}$ to find the height.

Q: What is the formula for the sine of an angle?

A: The formula for the sine of an angle is:

$$\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}$$

Where $x$ is the angle, and the opposite and hypotenuse are the sides of the triangle.

Q&A: Algebra and Functions

Q: What is a function, and how do I graph it?

A: A function is a relation between a set of inputs and a set of possible outputs. To graph a function, you need to plot the points on a coordinate plane and then connect them with a smooth curve.

Q: How do I find the domain and range of a function?

A: To find the domain and range of a function, you need to identify the set of possible inputs and outputs. You can use the graph of the function to determine the domain and range.

Q: What is the difference between a linear and a quadratic function?

A: A linear function is a function that can be written in the form $f(x) = mx + b$, where $m$ and $b$ are constants. A quadratic function is a function that can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Conclusion

In this article, we have answered some of the most frequently asked questions in mathematics, covering topics such as factoring, quadratic equations, geometry, trigonometry, algebra, and functions. We hope that this article has provided a helpful guide to these mathematical concepts and has inspired you to learn more about the world of mathematics.