Quadratic Equations, Inequalities and Functions

Chapter 11: Quadratic Equations, Inequalities and Functions

• Discuss different methods of solving quadratic equations
• Methods for graphing quadratic equations
• Applications of quadratic equations

Section 11.1: Solving Quadratic Equations using the Square Root Property

• Quadratic equations come in the form of ; standard form
• To be able to solve a quadratic equation, it must be in standard form

Zero Factor Property

• If two numbers have a product of zero , then at least one of the numbers must be zero.
• We use this property to solve equations after they have been factored

Example: Use the zero factor property to solve the given quadratic equation.

Square Root Property

• If k is a positive number and if , then

The solution can also be written as

• If the quadratic equation is of the form then the square root property can be used

Examples: Solve each equation. Write radicals in simplified form.

Section 11.2: Solving Quadratic Equations by Completing the Square

• A method for solving quadratic equations
• The goal is to factor the left side of the quadratic equation so that it is a perfect square
• The right side of the equation is a constant
• The square root property is the used to finish the solving of the equation
• We start with an equation in standard form
• The end result is of the form

Steps to Complete the Square

1. Make sure a is 1. If a is not 1, then perform the proper division
2. Write in the form ; variable terms on the left, constant term on the right
3. Complete the square using the formula

4. Add the value found in step 3 to both sides of the equation
5. Factor the left side as a perfect square, simplify the right side
6. Use the square root property to solve

Examples: Solve the given equations using completing the square.

Section 11.3: Solving Quadratic Equations by the Quadratic Formula

• Another method for solving quadratic equations
• Before the quadratic formula can be used the quadratic equation must be in standard form

Quadratic Formula

Examples: Use the quadratic formula to solve the given quadratic equations.

Discriminant

• A part of the quadratic formula;
• The discriminant can be used to determine the number and type of solutions a quadratic equation has
• The following table lays out the types of solutions

Examples: Find each discriminant. Use it to predict the number and type of solutions for each equation.

Example: Find k so that the equation will have exactly one rational solution.

Section 11.4: Equations in Quadratic Form

• We can solve quadratic equations by 4 different methods
 -Factoring
 -Square Root Property
 -Completing the Square
 -Quadratic Formula
• Some equations can be simplified down to quadratic form

Example: Solve the given equations

Examples: Solve the given application problems. Use the problem solving steps discussed previously.

1. In 1 ¾ hours Khe rows his boat 5 miles up the river and comes back. The speed of the current is 3 mph. How fast does Khe row?

2. Two chefs are preparing a banquet. One chef could prepare the banquet in 2 hours less time than the other. Together they can complete the job in 5 hours. How long would it take the faster chef working alone?

Section 11.5: Formulas and Further Applications
• This section illustrates the many uses of quadratic equations
• We also look at formulas

Examples: Solve the given formulas for the specified variable.

• Recall the Pythagorean Theorem,

Example: A ladder is leaning against a house. The distance from the bottom of the ladder to the house is 5 ft. The distance from the top of the ladder to the ground is 1 ft less than the length of the ladder. How long is the ladder?

Example: A ball is projected upward from the ground. Its distance in feet from the ground at t seconds is

a) At what time will the ball hit the ground?

b) At what time will the ball be 32 ft from the ground?

Section 11.6: Graphs of Quadratic Functions
• We now look at how to graph quadratic equations
• In this section we look at horizontal shifts and vertical shifts associated with parabolas
• We can also use a table of values to plot points and determine the graph of quadratic equations
• A quadratic function is of the form ; standard form
• A quadratic function can also be in the form of
• Functions of the form have a vertical shift of k units up if k is positive and k units down if k is negative
• Functions of the form have a horizontal shift of h units to the right if h is positive and h units to the left if h is negative
• Recall the ideas of domain and range; interval notation
• Recall the general shape of quadratic functions
 

General Principles
1. Graph the quadratic function defined by is a parabola with vertex
2. The graph opens up if a is positive and down if a is negative
3. The graph is wider than that of . The graph is narrower than that of

Examples: Graph the following functions. Use the ideas of vertical shift, horizontal shift and the general principles.

Section 11.7: More about Parabolas and Their Applications
• We use a similar set of principles when graphing quadratic equations in standard form
• To determine the vertex of the parabola , we use the following formula

General Principles of a Quadratic Function in Standard Form
1. Determine whether the graph opens up or down. If a>0 the parabola opens up. If a<0 the parabola opens down.
2. Find the vertex using the vertex formula.
3. Find the x and y intercepts. Recall that an x intercept is found when y is replaced with zero and the equation is solved for x. A y intercept is found when x is replaced with zero and the equation is solved for y; this usually involves factoring.
4. Graph the vertex and intercepts. Plot additional points as needed.

Example: Graph the given quadratic equation. Use the general principles as described above. Also, determine the domain and range.

• We can use the vertex formula to solve applications of quadratic equations
• Application problems which involve maximum or minimum areas require the use of the vertex formula

Example: Use the vertex formula to solve the following application problems.
1. A farmer has 100 ft of fencing. He wants to put a fence around the rectangular field next to a building. Find the maximum area he can enclose and the dimensions of the field when the area is maximized.

2. A toy rocket is launched from the ground so that its distance in feet above the ground after t seconds is. Find the maximum height it reaches and the number of seconds it takes to reach that height.