Problem 105 - College Algebra

Question

The formula $$ N=2 x^{2}+22 x+320 $$ models the number of inmates, \(N,\) in thousands, in U.S. state and federal prisons \(x\) years after 1980 . The graph of the formula is shown in a [0,20,1] by [0,1600,100] viewing rectangle at the top of the next column. Use the formula to solve Exercises 105-106. (graph cant copy) In which year were there 740 thousand inmates in U.S. state and federal prisons? Identify the solution as a point on the graph shown.

Step-by-Step Solution

Answer

There were 740 thousand inmates in U.S. state and federal prisons in the year 1986.

Step 1: Formulating the Equation

To find the year when there were 740 thousand inmates, replace \(N\) in the equation \(N=2x^2+22x+320\) with 740 and solve for \(x\). This gives us the equation \(740=2x^2+22x+320\).

Step 2: Setting the Equation to Zero

Rearrange the equation from Step 1 to set it to zero. Therefore, the equation will be \(2x^2+22x+320-740=0\), which simplifies to \(2x^2+22x-420=0\).

Step 3: Solving the Quadratic Equation

Now, solve the quadratic equation \(2x^2+22x-420=0\) for \(x\). We can solve it either by factoring, completing the square, or using the quadratic formula \((-b±√(b^2-4ac))/2a\). The solve gives \(x=6\) and \(x=-35\).

Step 4: Choosing the Correct Solution

Since \(x\) denotes years after 1980, \(x\) cannot be negative. Therefore, we discard \(x=-35\) and accept \(x=6\) as the solution. So the year was 6 years after 1980, which is 1986

Key Concepts

Quadratic Formula

When faced with a quadratic equation, the quadratic formula is a powerful tool to find its solutions. It is written as \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a\), \(b\), and \(c\) are the coefficients from a quadratic equation in the standard form \(ax^2 + bx + c = 0\). This method is universally applicable and especially useful when factoring is complex or impossible. Essentially, the quadratic formula simplifies the process of finding the roots of the equation by providing a straight-forward calculation. It deals neatly with both real and complex solutions, and is a surefire way to obtain the x-values where a parabola crosses the x-axis.

For instance, given the equation \(2x^2+22x-420=0\), applying the quadratic formula would allow us to calculate its roots. While the exercise at hand simplifies the equation, knowing how to utilize the formula with larger coefficients or when factoring is not practical is crucial for a deeper understanding of quadratic equations.

Factoring Quadratics

Factoring quadratics is another strategy to solve quadratic equations, where the goal is to decompose the quadratic into a product of binomials. The process involves finding two numbers that multiply to the product of the coefficient \(a\) and the constant term \(c\), and add up to the coefficient \(b\). These numbers are then used to break down the middle term and factor by grouping. However, factoring is not always possible, especially when dealing with prime numbers or in cases where the equation doesn’t factor neatly.

In the context of our problem, \(2x^2+22x-420=0\) can be factored if we look for two numbers that multiply to \(2 \times (-420) = -840\) and add up to 22. Factoring quadratics is a preferred method when it's feasible as it often provides a quick and intuitive solution to finding the zeros of the quadratic function, which represent the points where the graph intersects the x-axis.

Completing the Square

Completing the square is a technique used to solve quadratic equations by turning them into a perfect square trinomial, which can then be solved by taking the square root of both sides. This method involves dividing the coefficient of the \(x\) term by 2, squaring it, and adding this square to both sides of the equation to form \( (x+d)^2\), where \(d\) is the value obtained.

Although this method was not necessary for the provided exercise, understanding it is vital. In cases where the quadratic formula might be daunting due to complex coefficients, or when factoring is not an option, completing the square offers an alternative path to finding solutions. With this method, not only can we solve equations, but we can also gain insight into the vertex form of the quadratic, which helps when analyzing its graph.

Interpreting Graphs of Quadratic Functions

The graph of a quadratic function is a parabola, which can either open upwards or downwards. Interpreting these graphs allows us to visualize the nature of the solutions of the quadratic equation. On a graph, the roots of the equation correspond to the points where the parabola intersects the x-axis. The vertex, the highest or lowest point on the graph, gives the maximum or minimum value of the function, and the axis of symmetry cuts the parabola into two mirror images.

For the exercise given, identifying the year with 740 thousand inmates involved recognizing the correct point on the parabola's graph where the y-value (the number of inmates) equals 740. Every point on this graph correlates to a real-world scenario, namely the number of inmates for a given year. This visual approach is especially helpful for understanding the implications of quadratic models in real-world contexts, such as predicting trends or analyzing past data.