3.1.4 Evaluate The Expression: $ \frac{5.2 X-4.2 {x-2}}{2 X-2 {x-1}} }$3.2 Solve For { X$}$ If ${ 2^{-2x = \frac{1}{8} }$QUESTION 4Given The Equation: ${ x^2 + 2kx + 4x + 9k = 0 }$4.1 Write The

by ADMIN 202 views

Introduction

In mathematics, evaluating expressions and solving equations are fundamental concepts that form the basis of various mathematical operations. These concepts are essential in solving problems in algebra, geometry, and other branches of mathematics. In this article, we will delve into the world of evaluating expressions and solving equations, focusing on two specific problems: evaluating an expression and solving for a variable.

3.1.4 Evaluate the Expression

Problem Statement

Evaluate the expression: 5.2x−4.2x−22x−2x−1\frac{5.2^x-4.2^{x-2}}{2^x-2^{x-1}}

Solution

To evaluate this expression, we need to simplify the numerator and denominator separately.

Simplifying the Numerator

The numerator can be simplified as follows:

5.2x−4.2x−2=5.2x−4.2x⋅4.2−2=5.2x−4.2x⋅14.22=5.2x−4.2x⋅116.84=5.2x−4.2x16.84\begin{aligned} 5.2^x-4.2^{x-2} &= 5.2^x-4.2^x \cdot 4.2^{-2} \\ &= 5.2^x-4.2^x \cdot \frac{1}{4.2^2} \\ &= 5.2^x-4.2^x \cdot \frac{1}{16.84} \\ &= 5.2^x-\frac{4.2^x}{16.84} \end{aligned}

Simplifying the Denominator

The denominator can be simplified as follows:

2x−2x−1=2x−2x⋅2−1=2x−2x⋅12=2x−2x2=2x−2x2\begin{aligned} 2^x-2^{x-1} &= 2^x-2^x \cdot 2^{-1} \\ &= 2^x-2^x \cdot \frac{1}{2} \\ &= 2^x-\frac{2^x}{2} \\ &= 2^x-\frac{2^x}{2} \end{aligned}

Evaluating the Expression

Now that we have simplified the numerator and denominator, we can evaluate the expression:

5.2x−4.2x−22x−2x−1=5.2x−4.2x16.842x−2x2=5.2x−4.2x16.842x−2x2⋅16.8416.84=5.2x⋅16.84−4.2x2x⋅16.84−2x=87.088−4.2x32.68−2x\begin{aligned} \frac{5.2^x-4.2^{x-2}}{2^x-2^{x-1}} &= \frac{5.2^x-\frac{4.2^x}{16.84}}{2^x-\frac{2^x}{2}} \\ &= \frac{5.2^x-\frac{4.2^x}{16.84}}{2^x-\frac{2^x}{2}} \cdot \frac{16.84}{16.84} \\ &= \frac{5.2^x \cdot 16.84-4.2^x}{2^x \cdot 16.84-2^x} \\ &= \frac{87.088-4.2^x}{32.68-2^x} \end{aligned}

3.2 Solve for x

Problem Statement

Solve for xx if: 2−2x=182^{-2x} = \frac{1}{8}

Solution

To solve for xx, we need to isolate xx on one side of the equation.

Isolating x

We can start by rewriting the equation as follows:

2−2x=182−2x=2−3−2x=−3x=32\begin{aligned} 2^{-2x} &= \frac{1}{8} \\ 2^{-2x} &= 2^{-3} \\ -2x &= -3 \\ x &= \frac{3}{2} \end{aligned}

4.1 Write the Equation

Problem Statement

Given the equation: x2+2kx+4x+9k=0x^2 + 2kx + 4x + 9k = 0

Solution

To write the equation, we need to simplify it by combining like terms.

Simplifying the Equation

The equation can be simplified as follows:

x2+2kx+4x+9k=0x2+(2k+4)x+9k=0\begin{aligned} x^2 + 2kx + 4x + 9k &= 0 \\ x^2 + (2k+4)x + 9k &= 0 \end{aligned}

Conclusion

In this article, we have evaluated an expression and solved for a variable. We have also written an equation by simplifying it. These concepts are essential in solving problems in algebra, geometry, and other branches of mathematics. By following the steps outlined in this article, you can evaluate expressions and solve equations with confidence.

Discussion

  • What are some common mistakes to avoid when evaluating expressions and solving equations?
  • How can you use technology to evaluate expressions and solve equations?
  • What are some real-world applications of evaluating expressions and solving equations?

References

Additional Resources

Introduction

In our previous article, we explored the concepts of evaluating expressions and solving equations. In this article, we will delve into a Q&A format, providing answers to common questions and scenarios related to evaluating expressions and solving equations.

Q&A

Q: What is the difference between an expression and an equation?

A: An expression is a mathematical statement that contains variables, constants, and mathematical operations, but does not contain an equal sign (=). An equation, on the other hand, is a mathematical statement that contains an equal sign (=) and is used to solve for a variable.

Q: How do I evaluate an expression with multiple variables?

A: To evaluate an expression with multiple variables, you need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

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

A: A linear equation is an equation in which the highest power of the variable is 1. A quadratic equation, on the other hand, is an equation in which the highest power of the variable is 2.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

x=−b±b2−4ac2a\begin{aligned} x &= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \end{aligned}

where aa, bb, and cc are the coefficients of the quadratic equation.

Q: What is the difference between a system of linear equations and a system of quadratic equations?

A: A system of linear equations is a set of two or more linear equations that are solved simultaneously. A system of quadratic equations, on the other hand, is a set of two or more quadratic equations that are solved simultaneously.

Q: How do I solve a system of linear equations?

A: To solve a system of linear equations, you can use the following methods:

  1. Substitution Method: Substitute the expression for one variable from one equation into the other equation.
  2. Elimination Method: Add or subtract the equations to eliminate one variable.
  3. Graphing Method: Graph the equations on a coordinate plane and find the point of intersection.

Q: What is the difference between a rational expression and a rational equation?

A: A rational expression is a fraction that contains variables and constants in the numerator and denominator. A rational equation, on the other hand, is an equation that contains a rational expression.

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you can follow these steps:

  1. Factor the numerator and denominator.
  2. Cancel out any common factors.
  3. Simplify the resulting expression.

Q: What is the difference between a polynomial expression and a polynomial equation?

A: A polynomial expression is a mathematical statement that contains variables and constants, but does not contain an equal sign (=). A polynomial equation, on the other hand, is a mathematical statement that contains an equal sign (=) and is used to solve for a variable.

Q: How do I solve a polynomial equation?

A: To solve a polynomial equation, you can use the following methods:

  1. Factoring Method: Factor the polynomial expression and set each factor equal to zero.
  2. Quadratic Formula Method: Use the quadratic formula to solve the polynomial equation.
  3. Graphing Method: Graph the polynomial equation on a coordinate plane and find the x-intercepts.

Conclusion

In this article, we have provided answers to common questions and scenarios related to evaluating expressions and solving equations. By following the steps outlined in this article, you can evaluate expressions and solve equations with confidence.

Discussion

  • What are some common mistakes to avoid when evaluating expressions and solving equations?
  • How can you use technology to evaluate expressions and solve equations?
  • What are some real-world applications of evaluating expressions and solving equations?

References

Additional Resources