Question

If \(x>0\) and \(2 x^{2}+6 x=8,\) then the average (arithmetic mean) of \(x+2,2 x-1,\) and \(x+4\) is equal to which of the following? a. -2 b. 3 c. 3.5 d. 5 e. 7

Short answer

The average is 3.

Step-by-step solution

Simplify the given equation

Start with the equation given: \[2x^2 + 6x = 8\] Subtract 8 from both sides to set the equation to 0: \[2x^2 + 6x - 8 = 0\]

Solve the quadratic equation

Factorize the quadratic equation \[2x^2 + 6x - 8 = 0\] Divide the whole equation by 2 to simplify: \[x^2 + 3x - 4 = 0\] Now, factor it as: \[(x+4)(x-1) = 0\] So, the solutions are: \[x = -4\] and \[x = 1\] Since \(x > 0\), we take \(x = 1\).

Calculate the values of the expressions

First, substitute \(x = 1\) into the expressions: For \(x + 2\), \[1 + 2 = 3\] For \(2x - 1\), \[2(1) - 1 = 1\] For \(x + 4\), \[1 + 4 = 5\]

Find the average

Now, find the average (arithmetic mean) of the three values: \[ \text{Average} = \frac{(3 + 1 + 5)}{3} \] Calculate the sum: \[3 + 1 + 5 = 9\] Now divide by 3: \[\frac{9}{3} = 3\]

Key concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of the form \(ax^2 + bx + c = 0\). The highest degree of the variable, usually denoted as \(x\), is 2. Quadratics can be solved using various methods, such as factoring, completing the square, or applying the quadratic formula.
In our exercise, we start with the equation \(2x^2 + 6x = 8\). To transform it into standard quadratic form, we subtract 8 from both sides, resulting in \(2x^2 + 6x - 8 = 0\). This is now in the form \(ax^2 + bx + c = 0\).
To solve for \(x\), we proceed by factoring.

Arithmetic Mean

The arithmetic mean, often simply called the average, is a measure of central tendency. It's calculated by summing a list of values and then dividing by the number of values. Mathematically, for a set of numbers \(a_1, a_2, \, ... \, , a_n\), the arithmetic mean is calculated as: \[ \text{Arithmetic Mean} = \frac{a_1 + a_2 + ... + a_n}{n} \]
In our problem, we need the average of the expressions \(x + 2\), \(2x - 1\), and \(x + 4\). After solving for \(x\), we found these values to be 3, 1, and 5 respectively. Then, the average is calculated as: \[ \text{Average} = \frac{(3 + 1 + 5)}{3} = 3 \]

Solving for Variables

To solve for a variable means to find the value of the variable that makes the equation true. In the context of our quadratic equation \(2x^2 + 6x - 8 = 0\), we simplify and factorize it to find the values of \(x\). We factored the equation as: \[(x + 4)(x - 1) = 0\] This gives us two solutions: \(x = -4\) and \(x = 1\). However, since the problem specifies \(x > 0\), we select \(x = 1\). This value of \(x\) is then used to find the results of the expressions given in the problem.

Factoring Polynomials

Factoring polynomials involves expressing the polynomial as a product of its factors. For quadratic equations, we look for two binomials that multiply to form the original quadratic polynomial. In our problem, we simplify \(2x^2 + 6x - 8 = 0\) by dividing the entire equation by 2, getting \(x^2 + 3x - 4 = 0\). Next, we factor it into: \[(x + 4)(x - 1) = 0\] This tells us that the values of \(x\) that satisfy the equation are -4 and 1. We use factoring to break down complex polynomials into simpler binomials, making it easier to solve for variables.