Solving a Quadratic Inequality for an Object in Motion
Quadratic equations can be used to represent the path of objects as they go up in the air, and come back down due to the force of gravity. Imagine a ball being thrown straight up into the air. For the first few seconds, the height of the ball will be increasing, but eventually, it will reach a maximum height and then begin to fall back down, eventually hitting the group. In relation to time, the graph might look something like this:
The graph of this curve can be represented by the following equation:
y = -16x2 + 80x + 6
The coefficients in this equation have important meanings within this context. -16x2 comes from the force of gravity on Earth, 80x represents the velocity of the ball, and 6 represents the height of the player hitting the ball.
We can use this equation to answer some questions about the height of the ball, and how long it is in the air. Suppose we want to figure out when the ball is more than 70 feet in the air. We will use a quadratic inequality to represent this situation.
70 < -16x2 + 80x + 6
We use the < symbol to show that the height of the ball (represented by the quadratic equation) has a greater value than 70. We could have alternatively written our inequality with 70 in the right side, where the inequality symbol would be >
To solve quadratic inequalities:
Our first step is to write as an equation set equal to zero. To do this, we'll need to move the constant term to the other side:
To solve this equation, we can use any method, although factoring and the quadratic formula are the most common. Factoring tends to involve fewer calculations, but it takes too long to identify such integers, it is probably best to use the quadratic formula. Here, we are going to solve by factoring, but first by factoring out a common factor between all coefficients.
We use these two solutions to form intervals on the number line. Since we have two solutions, this breaks the number line into three sections:
Next, we choose a value that lies within each interval, and plug that value in to the inequality, to see if it yields a true statement. It is helpful to use the inequality with zero on the one side, so we can simply compare the values to zero. We'll choose the test points: 0, 2, and 5.
You might prefer to use the inequality written in factored form, especially if the other side of the inequality is zero. This is because you can simply analyze the signs of each factor (positive or negative) to determine if the value of the quadratic will be positive or negative. From there, you can easily compare to zero without actually calculating exact values.
This means that the ball is above 70 feet between 1 and 4 seconds after being launched into the air.
Quadratic Inequalities with No Solution
Next, we would like to know when the ball is above 120 feet. To answer this question, we follow the same steps as in our previous example, but with a different inequality to represent this new scenario.
Our inequality is: 120 < -16x2 + 80x + 6
Writing as an equation set equal to zero, we must solve 0 = –16x2 + 80x – 114. We will solve this using the quadratic formula.
We cannot work any further, because we have a negative value underneath the square root. There is a special term for this value when working with the quadratic formula. We call this the discriminant. If the discriminant is negative, there is no real solution to the equation, because every real number square leads to a positive value.
Since there is no real solution, we know that the ball will never even reach 120 feet in the air.