a. Solve by completing the square.
b. Solve using the quadratic formula.
Example: Given the quadratic equation:
a. Solve by factoring.
b. Solve using the quadratic formula.
Example: Graph. Label the x- and y-intercepts as well as the vertex.
Projectile Problem: An object is launched from an 80-foot tower with an initial speed of 75 feet per second. At what times is the object 120 feet high? Use the following function:
Geometry Problem: The height of a triangle is 2 centimeters more than 3 times the base. If the area is 28 square centimeters then what is the length of the base and height?
Example: Find a quadratic equation with integer coefficients and the following solution set.
Example: The product of two consecutive odd positive integers is eleven less than 10 times the larger. Find the integers.
Example: The length of a rectangle is 6 centimeters longer than twice its width. If the area is 140 square centimeters, find the dimensions of the rectangle.
Example: The cost of a particular car in dollars can be approximated by the function:
Here x represents the age of the car in years.
a. Use the graph to determine the cost of the car when it was new?
b. How old will the car be when it reaches its minimum cost?
c. How much is this car worth when it reaches 5 years old?
In addition, we will revisit function notation and apply the techniques in this section to quadratic functions.
The above zero factor property is the key to solving quadratic equations by factoring. So far we have been solving linear equations, which usually had only one solution. We will see that quadratic equations have up to two solutions.
Solve:
Step 1: Obtain zero on one side and then factor.
Step 2: Set each factor equal to zero.
Step 3: Solve each of the resulting equations.
This technique requires the zero factor property to work so make sure the quadratic is set equal to zero before factoring in step 1.
Tip: You can always see if you solved correctly by checking your answers. On an exam it is useful to know if got the correct solutions or not.