Instructions: Solve The Following Problems. Show Your Solutions When Needed.1. Evaluate: $2\left(5^2\right) - 3(4$\]2. Factor Completely: $x^2 - 9x + 18$3. Solve For $x$: $3x + 7 = 2x + 12$4. Find The Slope Of The

by ADMIN 214 views

In this article, we will guide you through the step-by-step process of solving various mathematical problems. We will cover four different types of problems, including evaluating expressions, factoring quadratic equations, solving linear equations, and finding the slope of a line.

Problem 1: Evaluate the Expression

Evaluating Expressions

To evaluate an expression, we 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.

Let's evaluate the expression: 2(52)−3(4)2\left(5^2\right) - 3(4)

Step 1: Evaluate the Exponential Expression

First, we need to evaluate the exponential expression 525^2. This means we need to raise 5 to the power of 2.

52=5×5=255^2 = 5 \times 5 = 25

Step 2: Substitute the Value into the Expression

Now that we have evaluated the exponential expression, we can substitute the value into the original expression.

2(52)−3(4)=2(25)−3(4)2\left(5^2\right) - 3(4) = 2(25) - 3(4)

Step 3: Multiply the Numbers

Next, we need to multiply the numbers inside the parentheses.

2(25)=502(25) = 50 3(4)=123(4) = 12

Step 4: Subtract the Numbers

Finally, we need to subtract the two numbers.

50−12=3850 - 12 = 38

Therefore, the value of the expression 2(52)−3(4)2\left(5^2\right) - 3(4) is 38.

Problem 2: Factor the Quadratic Equation

Factoring Quadratic Equations

To factor a quadratic equation, we need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.

Let's factor the quadratic equation: x2−9x+18x^2 - 9x + 18

Step 1: Find the Factors of the Constant Term

First, we need to find the factors of the constant term 18.

The factors of 18 are: 1, 2, 3, 6, 9, 18

Step 2: Find the Pair of Factors that Add Up to the Coefficient of the Linear Term

Next, we need to find the pair of factors that add up to the coefficient of the linear term -9.

The pair of factors that add up to -9 are: -3 and -6

Step 3: Write the Factored Form of the Quadratic Equation

Now that we have found the pair of factors, we can write the factored form of the quadratic equation.

x2−9x+18=(x−3)(x−6)x^2 - 9x + 18 = (x - 3)(x - 6)

Therefore, the factored form of the quadratic equation x2−9x+18x^2 - 9x + 18 is (x−3)(x−6)(x - 3)(x - 6).

Problem 3: Solve the Linear Equation

Solving Linear Equations

To solve a linear equation, we need to isolate the variable on one side of the equation.

Let's solve the linear equation: 3x+7=2x+123x + 7 = 2x + 12

Step 1: Subtract the Linear Term from Both Sides

First, we need to subtract the linear term 2x from both sides of the equation.

3x−2x+7=2x−2x+123x - 2x + 7 = 2x - 2x + 12

Step 2: Simplify the Equation

Next, we need to simplify the equation.

x+7=12x + 7 = 12

Step 3: Subtract 7 from Both Sides

Now, we need to subtract 7 from both sides of the equation.

x+7−7=12−7x + 7 - 7 = 12 - 7

Step 4: Simplify the Equation

Finally, we need to simplify the equation.

x=5x = 5

Therefore, the solution to the linear equation 3x+7=2x+123x + 7 = 2x + 12 is x = 5.

Problem 4: Find the Slope of the Line

Finding the Slope of a Line

To find the slope of a line, we need to use the formula:

m = (y2 - y1) / (x2 - x1)

Let's find the slope of the line that passes through the points (2, 3) and (4, 5).

Step 1: Identify the Coordinates of the Points

First, we need to identify the coordinates of the points.

Point 1: (2, 3) Point 2: (4, 5)

Step 2: Plug the Values into the Formula

Next, we need to plug the values into the formula.

m = (5 - 3) / (4 - 2)

Step 3: Simplify the Equation

Now, we need to simplify the equation.

m = 2 / 2 m = 1

Therefore, the slope of the line that passes through the points (2, 3) and (4, 5) is 1.

Conclusion

In this article, we will answer some of the most frequently asked questions about mathematical problem-solving. Whether you are a student, teacher, or simply someone who wants to improve their math skills, this article is for you.

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is:

  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: How do I factor a quadratic equation?

A: To factor a quadratic equation, we need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. We can use the following steps to factor a quadratic equation:

  1. Find the factors of the constant term.
  2. Find the pair of factors that add up to the coefficient of the linear term.
  3. Write the factored form of the quadratic equation.

Q: How do I solve a linear equation?

A: To solve a linear equation, we need to isolate the variable on one side of the equation. We can use the following steps to solve a linear equation:

  1. Subtract the linear term from both sides of the equation.
  2. Simplify the equation.
  3. Add or subtract the same value from both sides of the equation to isolate the variable.
  4. Simplify the equation.

Q: What is the slope of a line?

A: The slope of a line is a measure of how steep the line is. It is calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line.

Q: How do I find the slope of a line?

A: To find the slope of a line, we need to use the formula:

m = (y2 - y1) / (x2 - x1)

We can use the following steps to find the slope of a line:

  1. Identify the coordinates of two points on the line.
  2. Plug the values into the formula.
  3. Simplify the equation.

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 is an equation in which the highest power of the variable is 2.

Q: How do I graph a linear equation?

A: To graph a linear equation, we need to find two points on the line and plot them on a coordinate plane. We can then draw a line through the two points to represent the linear equation.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, we need to find the x-intercepts of the equation and plot them on a coordinate plane. We can then draw a parabola through the x-intercepts to represent the quadratic equation.

Conclusion

In this article, we have answered some of the most frequently asked questions about mathematical problem-solving. Whether you are a student, teacher, or simply someone who wants to improve their math skills, this article is for you. By following the instructions and examples provided in this article, you should be able to solve mathematical problems with ease.

Additional Resources

  • For more information on mathematical problem-solving, please visit our website at [insert website URL].
  • For additional practice problems and exercises, please visit our online math resource at [insert website URL].
  • For one-on-one tutoring or online math classes, please contact us at [insert contact information].

We hope this article has been helpful in answering your questions about mathematical problem-solving. If you have any further questions or need additional assistance, please don't hesitate to contact us.