Quadratic Equations

An introduction to

Quadratic Equations

Definition: A quadratic function can be expressed in the form: where , a,b, and c are real numbers. The graph of a quadratic function is a called parabola.

Example 1: Investigate the effects of various values for a,b, and c on the graph of the function, . If available, you can use a graphing calculator, but in the absence of a calculator, much can be learned by starting with a few special quadratic functions, using a table, and carefully changing the values for a,b, and c. Be prepared to describe the functions you graphed and your conjectures based on the graphs you studied.

Example 2: The equation describes the path of a basketball after it is tossed.

Example 3: A quadratic function can also be written in the form, which sometimes is more useful. Investigated the effects of various values for a, h, and k in the equation . How does your analysis of this form differ from what you learned with Example 1?

Example 4: The observation deck of the Tower of the Americas in San Antonio, Texas, is 622 feet above the ground. You can use the formula to find the height (in feet) of an object t seconds after it is dropped from the observation deck.

a)Find at least 4 order pairs to plot and then connect the points with a smooth curve. Verify your graph using your graphing calculator.

b)About how long does it take an object dropped from the observation deck to hit the ground?

Example 5: Different ways of thinking about a problem can lead to different methods for solving it. For example, finding the x-intercepts of the graph of is the same as solving the equation. The solutions to are called the roots of the equation.

Find the x-intercepts of the graph of i.e., solve.

Example 6: Find the x-intercepts of the graph of i.e., solve.

Quadratic formula

Another method of solving where , a,b, and c are real numbers is to use the quadratic formula: .

Notice that if

then there are no real solutions to the equation,

, then there is a unique solution, and

, then there are 2 solutions.

Where does the quadratic formula come from?

Example 7:Let’s first solve using a mix of algebra and geometry. Then we will think about solving the abstract quadratic equation, .

Completing the square: Consider for a moment just the expression

THEREFORE

Example 8: Given where , a,b, and c are real numbers we will begin by factoring out the a and then completing the square.

Example 9:

a)Use the quadratic formula to find the x-intercepts of the following quadratic equation

b)Recall from Example 1 (page 1 of this section),describes the path of a basketball after it is tossed. Find the exact horizontal distance traveled when the ball hits the ground?

c)From Example 1, we also wish to determine the exact value of the maximum height of the ball. Hint: First find the horizontal distance by using the values of the x-intercepts and the symmetry of the parabola. Then use this information to find the maximum height of the basketball.

d)Recall from Example 2 (page 2 of this section), describes the height (in feet) of an object t seconds after it is dropped from the observation deck. How long does it take an object dropped from the observation deck to hit the ground?

Quadratic Equations – page 1