Chapter 3

Quadratic Fns & Eqns;Inequalities

Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Sec 3.2

Quadratic Equations, Functions, Zeros, and Models

Quadratic Equations

A quadratic equation is an equation that can be written in the form:

ax2 + bx + c = 0; a 0, where a, b, and c are real numbers.

A quadratic equation in this form is in standard form.

Quadratic Functions

A quadratic function is of the form:

f (x) = ax2 + bx + c, a 0, where a, b, and c are real numbers.

The zeros of a quadratic function are the solutions of the associated equation; ax2 + bx + c = 0.

Quadratic functions can have real (curve touches horiz. axis), or imaginary (curve never touches horiz. axis) zeros. They must have either 2 Re or 2 Imag zeros.

Equation-Solving Principle #1

The “Principle of Zero Products” or Zero-Product Rule:

If a·b = 0, then the only way that can happen is either:a = 0 or b = 0 or both a,b = 0,

Equation-Solving Principle #2

The Principle of Square Roots: If x2 = k, then either

x k or x k .

Solving Quadratic Equations

Solve: 2x2 x = 3.

2x2 x 3

2x2 x 3 0

x 1 2x 3 0

x 1 or x 3

2

IMPORTANT: ALWAYS verify ALL solutions!

Example

Solve: 2x2 10 = 0.

5 .The solutions are and5

Completing the Square

1. Isolate the terms with variables on one side of the equation and constant on the other in descending order.

2. Divide all terms by the leading coefficient if that coefficient is not 1.

3. Complete the square by making the left side a “perfect square”.

4. Take the square root of both sides.5. Solve for the variable.

To solve a quadratic equation by completing the square:

Example

Solve by CTS: 2x2 1 = 3x.

2x2 3x 1

The solutions are 3 17

4.

Quadratic Formula

The solutions of ax2 + bx + c = 0, a 0, are given by

This formula can be used to solve any quadratic equation.

x b b2 4ac

2a.

The Discriminant : “D”

Using the quadratic formula, you will find, you find the value of (b2 4ac), which can be pos, neg, or zero.

This value is called the discriminant.

For ax2 + bx + c = 0, where a, b, and c are real numbers:

b2 4ac = 0 One real-number solution; b2 4ac > 0 Two different real-number solutions; b2 4ac < 0 Two different complex conjugates solutions.

Solving Quadratic Equations

Solve: 3x2 + 2x = 7

1. Use the Discriminant to determine the nature of the solutions.

2. Solve using the Quadratic Formula (use 3-decimal accuracy for approximations)

...

The exact solutions are:

The approximate solutions are: –1.897 and 1.230

Now, do this on the calculator….

1 22

3 or

1 22

3

Equations Reducible to Quadratic

Some equations can be treated as quadratic, even if they are not, provided that we make a suitable substitution.

Solve: x4 5x2 + 4 = 0 Hint: x4 = (x2)2, so substitute “u” for x2

Equations like this are said to be reducible to quadratic, or quadratic in form.

The solutions are : x = ±1, ±2 (VERIFY!)

Quadratic Applications: Gravity

The Petronas Towers in Kuala Lumpur, Malaysia stand 1482 ft above ground level.

How long would it take an object, dropped from the top of the tower, to reach the ground?

Hint: The quadratic formula: is used to

approximate the distance “s” (ft), that an object falls in “t” (sec) where the accel due to gravity: a=32 fps/s

2

2

1ats

9.624 sec