Calculating X From Mean And Solving Quadratic Equations By Completing The Square

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1. Introduction

In this article, we will explore two distinct mathematical problems. First, we will delve into the concept of the mean and how to calculate an unknown value within a dataset given its mean. Specifically, we will determine the value of x when the mean of a set of numbers, including x itself, is equal to x. Second, we will tackle a quadratic equation and employ the completing the square method to find its solutions, rounding our answers to one decimal place. These exercises provide valuable practice in fundamental algebraic techniques and problem-solving strategies. Understanding these concepts is crucial for further studies in mathematics and its applications in various fields.

2. Calculating x when the Mean is Known

2.1 Understanding the Mean

To calculate x when the mean of 30, x, 12, 40, and 10 is equal to x, we must first define the mean. The mean, often referred to as the average, is a measure of central tendency found by summing all the values in a dataset and dividing by the number of values. In this case, our dataset consists of five numbers: 30, x, 12, 40, and 10. The formula for the mean is:

Mean = (Sum of values) / (Number of values)

2.2 Setting up the Equation

We are given that the mean of these five numbers is equal to x. Therefore, we can set up the following equation:

x = (30 + x + 12 + 40 + 10) / 5

This equation represents the core of our problem. It states that the unknown value x is equal to the sum of all the values in the dataset divided by the number of values, which is 5.

2.3 Solving for x

Now, let's solve this equation for x. The first step is to multiply both sides of the equation by 5 to eliminate the fraction:

5x = 30 + x + 12 + 40 + 10

Next, we simplify the right side of the equation by adding the constants:

5x = x + 92

Now, we need to isolate x on one side of the equation. Subtract x from both sides:

5x - x = 92

4x = 92

Finally, divide both sides by 4 to solve for x:

x = 92 / 4

x = 23

Therefore, the value of x that satisfies the given condition is 23. This means that if we substitute 23 for x in our original dataset, the mean of the numbers will indeed be 23.

2.4 Verification

To ensure our solution is correct, we can substitute x = 23 back into the original equation and verify that the mean is indeed 23:

Mean = (30 + 23 + 12 + 40 + 10) / 5

Mean = 115 / 5

Mean = 23

This confirms that our solution x = 23 is correct. The mean of the numbers 30, 23, 12, 40, and 10 is indeed 23.

3. Solving Quadratic Equations by Completing the Square

3.1 Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:

ax2 + bx + c = 0

where a, b, and c are constants, and a ≠ 0. Quadratic equations have numerous applications in various fields, including physics, engineering, and economics. There are several methods to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. In this section, we will focus on the method of completing the square.

3.2 The Method of Completing the Square

Completing the square is a technique used to rewrite a quadratic equation in a form that allows us to easily find its solutions. The main idea is to transform the quadratic expression into a perfect square trinomial, which can then be factored as (x + h)2 or (x - h)2, where h is a constant. Here are the steps involved in completing the square:

  1. Divide by a: If a ≠ 1, divide the entire equation by a to make the coefficient of x2 equal to 1.
  2. Move the constant term: Move the constant term (c) to the right side of the equation.
  3. Complete the square: Take half of the coefficient of the x term (which is b), square it, and add it to both sides of the equation. This step creates a perfect square trinomial on the left side.
  4. Factor the perfect square trinomial: Factor the left side as a squared binomial, which will be in the form (x + h)2 or (x - h)2.
  5. Take the square root: Take the square root of both sides of the equation.
  6. Solve for x: Solve the resulting equation for x.

3.3 Solving x^2 - 8x + 3 = 0 by Completing the Square

Let's apply the method of completing the square to solve the equation:

x2 - 8x + 3 = 0

  1. Divide by a: In this case, a = 1, so we don't need to divide.

  2. Move the constant term: Subtract 3 from both sides of the equation:

    x2 - 8x = -3

  3. Complete the square: The coefficient of the x term is -8. Half of -8 is -4, and (-4)2 is 16. Add 16 to both sides of the equation:

    x2 - 8x + 16 = -3 + 16

    x2 - 8x + 16 = 13

  4. Factor the perfect square trinomial: The left side is now a perfect square trinomial, which can be factored as:

    (x - 4)2 = 13

  5. Take the square root: Take the square root of both sides of the equation:

    √((x - 4)2) = ±√13

    x - 4 = ±√13

  6. Solve for x: Add 4 to both sides of the equation:

    x = 4 ± √13

3.4 Approximating the Solutions

We now have two solutions for x:

x1 = 4 + √13

x2 = 4 - √13

To approximate these solutions to one decimal place, we need to find the approximate value of √13. The square root of 13 is approximately 3.60555. Therefore:

x1 ≈ 4 + 3.60555 ≈ 7.60555

x2 ≈ 4 - 3.60555 ≈ 0.39445

Rounding these values to one decimal place, we get:

x1 ≈ 7.6

x2 ≈ 0.4

Thus, the solutions to the equation x2 - 8x + 3 = 0, correct to one decimal place, are approximately 7.6 and 0.4.

4. Conclusion

In this article, we have successfully solved two mathematical problems. First, we calculated the value of x when the mean of a set of numbers, including x, was equal to x. We found that x = 23. Second, we solved the quadratic equation x2 - 8x + 3 = 0 by completing the square and approximated the solutions to one decimal place, obtaining x ≈ 7.6 and x ≈ 0.4. These exercises demonstrate the importance of understanding fundamental algebraic concepts and techniques for problem-solving in mathematics.