G9 Math Q1- Week 1 Intro of Quadratic Equation.ppt

Illustrations of quadratic Equations

1. QUADRATIC
EQUATIONS

2. OBJECTIVE:
•illustrates quadratic equations.
(M9AL-Ia-1)

3. A quadratic equation is
an equation containing the second power of an
unknown but no higher power.  
The equation x
2
 - 5x+6 = 0
is a quadratic equation.  A quadratic
equation has two roots, both of which satisfy
the equation.  

4. The two roots of the quadratic
equation 

x
2
 - 5x+6 = 0 are x = 2 and x = 3.   
Substituting either of these values
for x in the equation makes it true.

5. The standard form of a quadratic
equation is the following:

6. •The a represents the numerical coefficient
of x
2
 , b represents the numerical coefficient of x, and
c represents the constant numerical term.  
•One or both of the last two numerical coefficients maybe
 zero.   The  numerical  coefficient  a  cannot  be  zero 
• If  b=0,  then  the  quadratic  equation  is termed a
"pure" quadratic equation.
• If the equation contains both an x and  x
2
 term, then it is a
complete  quadratic  equation.   

7. Some examples of quadratic
equations include:
3x
2
 +
9x - 2 = 0
6x
2
 +
11x = 7
4x
2
 =
13

8. •The name quadratic comes
from "quad" meaning square,
because the variable is squared
 (in other words x
2
).

9. SOLVING QUADRATIC EQUATIONS
The four axioms used in solving linear
equations are also used in solving quadratic
equations. However,  there  are  certain
 additional  
rules  used  when  solving  quadratic
equations.   

10. There  are three different techniques used for
solving quadratic equations:
1. taking the square root,
2. factoring
3. Using the Quadratic Formula.  
Of these three techniques, only the
Quadratic Formula will solve all quadratic
equations.   The other two techniques can
be used only in certain cases.   To determine
which technique can be used,
the equation must be written in general
form: ax
2
 + bx + c = 0

11. To determine which technique can be used, the equation must be
written in general form: ax
2
 + bx + c = 0
•If the equation is a pure quadratic equation (b=0) it can be solved by
taking the square root.  
Ex. 4x
2
-1 = 0, 4x
2
= +1, x
2
= 1/4, taking the square root of ¼
we get the two solutions x= +1/2 and x= -1/2
•If the numerical constant c is zero,  the equation can be solved
by factoring.
Ex. 4x
2
– 3x = 0 , x(4x- 3)=0, for the zero – factor property[1] x = 0
, 4x-3 =0, so the two solutions are x=0 and x=+3/4
Certain other equations can also be solved by factoring and applying the
zero – factor property.
Ex. x
2
 + 5x+6 = 0, if we factor we have
(x+3) (x+2)= 0 then x+3 =0 , x+2 =0 so the two solutions are x= -3
and x = -2

12. The solution(s) to a quadratic equation can always be calculated using
the Quadratic Formula:

13. •The "±" means you need to do a plus AND a minus, and
therefore there are normally TWO solutions ! You can try to
solve any quadratic equation by using the quadratic formula.
•The blue part =
Δ
b
2
 - 4ac is called the "discriminant",
because it can "discriminate" between the possible types of
answer. If it is positive, you will get two real solutions, if it is
zero you get just ONE solution, and if it is negative you get 
no real solutions.
•This formula gives two solutions, although the two
solutions may be the same number. (When solving
any polynomial equation of degree n, there are at
most n solutions to that equation.)

14. GRAPHING QUADRATIC
EQUATIONS
•When you graph a quadratic equation, you get a 
parabola, and the solutions to the quadratic equation
represent where the parabola crosses the x-axis.
•Every quadratic equation has at most two solutions,
but for some equations, the two solutions are the
same number, and for others, there is no solution on
the number line (because it would involve the square
root of a negative number).

15. •If the discriminant >0
Δ
, the quadratic equation has
2 distinct solutions and there will be two distinct  x-
intercepts.
•If the discriminant = 0
Δ
, the quadratic equation has
1 solution and there will be just one x-intercept.
•If the discriminant <0
Δ
, the quadratic equation has
no solutions and there will be no x-intercepts.

16. The
 
green
parabola
 
has 2 x-intercepts. Its
corresponding quadratic equation has 2 distinct
solutions (x=1 and x=4). This happens when the
discrimant Δ>0
The
 
yellow
parabola
 
has one x-intercept. Its
corresponding quadratic equation has 1 solution (x=-3).
Δ = 0
The
 
purple
parabola
 
has no x-intercepts. Its
corresponding quadratic equation has no
solutions. Δ < 0