# Quadratic Equations, Inequalities and Functions

# 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

Discriminant |
Number and Type of Solutions |

Positive, and the square of an integer | 2 rational solutions |

Positive, but not the square of an integer | 2 irrational solutions |

Zero | 1 rational solution |

Negative | 2 non-real complex 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.

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