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Section 1.5Quadratic Equations
2
Definition of a Quadratic Equation
A quadratic equation in x is an equation that can be written
in the general form
0,
where a, b, and c are real numbers, with a 0. A quadratic
equation in x is a
ax bx x
lso called a second-degree polynomial
equation in x.
Solving Quadratic Equations
by Factoring
The Zero-Product Principle
If the product of two algebraic expressions
is zero, then at least one of the factors is
equal to zero.
If AB=0, then A=0 or B=0
2
Solving a Quadratic Equation by Factoring
1. If necessary, rewrite the equation in the general
form ax +bx+c=0, moving all terms to one side,
thereby obtaining zero on the other side.
2. Factor completely.
3. Apply the zero-product principle, setting each factor
containing a variable equal to zero.
4. Solve the equations in step 3.
5. Check the solutions in the original equation
Example
2 5 6 0x x Solve by factoring:
Example
Solve by factoring:
2 3 40x x
Example
Solve by factoring:
22 11 15 0x x
Graphing CalculatorThe real solutions of a quadratic equation ax2+bx+c=0 correspond to the x-intercepts of the graph. The U shaped graph shown below has two x intercepts. When y=0, the value(s) of x will be the solution to the equation. Since y=0 these are called the zeros of the function.
Solving Polynomial Equations using the Graphing Calculator
Repeat this process for each x intercept.
By pressing 2nd Trace to get Calc, then the #2,you get the zeros. It will ask you for left and right bounds, and then a guess. For left and right bounds move the blinking cursor (using the arrow keys-cursor keys) to the left and press enter. Then move the cursor to the right of the x intercept and press enter. Press enter when asked to guess. Then you get the zeros or solution.
Solving Quadratic Equations
by the Square Root Property
2
2
2
The Square Root Property
If u is an algebraic expression and d is a
nonzero real number, then u =d has
exactly two solutions.
If u , then u= d or u=- d.
Equivalently,
If u , then u= d.
d
d
Example
Solve by the square root property:
24 44 0x
Example
Solve by the square root property:
22 7x
Completing the Square
22
22
2
Completing the Square
bIf x bx is a binomial, then by adding ,
2
which is the square of half the coefficient of x,
a perfect square trinomial will result. That is,
b bx bx+
2 2
x +8x
x
2
22
8 add
2
-7x 7x add
2
Why we call this completing the square.
Example
What term should be added to each binomial so that it becomes a perfect square trinomial? Write and factor the trinomial.
2
2
10
9
x x
x x
Example
Solve by Completing the Square:
2 8 10 0x x
Example
Solve by Completing the Square:
2 14 29 0x x
Solving Quadratic Equations
Using the Quadratic Formula
2
2
The Quadratic Formula
The solutions of a quadratic equation in general
form ax bx+c=0, with a 0, are given by the
quadratic formula
-b b 4x=
2
ac
a
Example
Solve by using the Quadratic Formula:
2
2
6 30 0
2 5 8 0
x x
x x
The Discriminant
Example
Compute the discriminant and determine the number and type of solutions:
2
2
2
2 3 7 0
5 4 0
6 1 0
x x
x x
x x
Determining Which
Method to Use
Applications
The Pythagorean Theorem
The sum of the squares of the
lengths of the legs of a right
triangle equals the square of the
length of the hypotenuse.
If the legs have lengths a and b,
and the hypotenuse has len2 2 2
gth c,
then a b c
Example
A machine produces open boxes using square sheets of metal. The
figure illustrates that the machine cuts equal sized squares measuring
2 inches on a side from the corners, and then shapes the metal into an
open box. Write the equation for the volume of this box. If the volume is
50 cubic inches, what is the length of the side of the original metal.
Example
A 42 inch television is a television whose screen’s diagonal length is 42 inches. If a television’s screen height is 26 inches, find the width of the television screen
42 inches
26 inches
(a)
(b)
(c)
(d)
2
Solve by the square root property.
x-4 15
4 15
4 15
19
4 15
(a)
(b)
(c)
(d)
2
Solve by completing the square.
x 12 3 0x
4 39
6 33
6 33
12 39
