G9 Math Q1- Week 1- Solves Quadratic Equation.ppt

it is about how to solve quadratic equations

1. QUADRATIC EQUATIONS

2. PERFORMANCE
STANDARD
The learners shall be able to
keenly observe the process
or steps on how to solve
quadratic equations.

3. LEARNING
COMPETENCY
The learner will be able to solve quadratic
equation by: (a) extracting square roots; (b)
factoring; (c) completing the square; (d) using
quadratic formula

4. At the end of the lesson,students will be
able to:
solves quadratic equations by:
(a) extracting square roots;
(b) factoring;
(c) completing the square; and (d) using the
quadratic formula. (M9AL-Ia-b-1)

5. QUADRATIC EQUATIONS ARE
WRITTEN IN THE FORM AX
2
+ BX
+ C = 0, WHERE A ≠ 0.

6. METHODS USED TO
SOLVE QUADRATIC EQUATIONS
1. Graphing
2. Factoring
3. Square Root Property
4. Completing the Square
5. Quadratic Formula

7. WHY SO MANY METHODS?
- Some methods will not work for
all equations.
- Variety is the spice of life.
- Some equations are much
easier to solve using a
particular method.

8. GRAPHING
Graphing to solve quadratic equations does not always
produce an accurate result.
If the solutions to the quadratic equation are irrational or
complex, there is no way to tell what the exact solutions are
by looking at a graph.
Graphing is very useful when solving contextual problems
involving quadratic equations.

9. GRAPHING (EXAMPLE 1)
y = x
2
– 4x – 5
Solutions are
-1 and 5

10. GRAPHING (EXAMPLE 2)
y = x
2
– 4x + 7
Solutions are
2 3i

11. GRAPHING (EXAMPLE 3)
y = 3x
2
+ 7x – 1
Solutions are
7 61
6
 

12. FACTORING
Factoring is typically one of the easiest and quickest ways
to solve quadratic equations;
however,
not all quadratic polynomials can be factored.
This means that factoring will not work to solve many
quadratic equations.

13. Solving Quadratic Equations Using
Factoring:
 
To Solve a Quadratic Using Factoring:
1)Put the quadratic equation into standard
form (above).
2)Factor the quadratic expression.
3)Set each factor equal to zero.
4)Solve each equation.
5)Check each root in the original
equation.

14. FACTORING (EXAMPLES)
Example 1
x
2
– 2x – 24 = 0
(x + 4)(x – 6) = 0
x + 4 = 0 x – 6 = 0
x = –4 x = 6
Example 2
x
2
– 8x + 11 = 0
x
2
– 8x + 11 is prime;
therefore, another method
must be used to solve this
equation.

15. SQUARE ROOT PROPERTY
This method is also relatively quick and easy;
however,
it only works for equations in which the quadratic
polynomial is written in the following form.
x
2
= n or (x + c)
2
= n

16. SQUARE ROOT PROPERTY (EXAMPLES)
Example 1Example 2
x
2
= 49 (x + 3)
2
= 25


x = ± 7 x + 3 = ± 5
x + 3 = 5 x + 3 = –5
x = 2 x = –8
2
49x 2
( 3) 25x 
Example 3
x
2
– 5x + 11 = 0
This equation is
not written in the
correct form to
use this method.

17. COMPLETING THE SQUARE
This method will work to solve ALL quadratic equations;
however,
it is “messy” to solve quadratic equations by completing the
square if a ≠ 1 and/or b is an odd number.
Completing the square is a great choice for solving
quadratic equations if a = 1 and b is an even number.

18. COMPLETING THE SQUARE (EXAMPLES
Example 1
a = 1, b is even
x
2
– 6x + 13 = 0
x
2
– 6x + 9 = –13 + 9
(x – 3)
2
= –4
x – 3 = ± 2i
x = 3 ± 2i
Example 2
a ≠ 1, b is not even
3x
2
– 5x + 2 = 0
25 2
0
3 3
x x  
25 25 2 25
3 36 3 36
x x   
2
5 1
6 36
x
 
 
 
 
 
5 1
6 6
x 
5 1
6 6
x 
5 1
6 6
x 
OR
x = 1 OR x =
⅔

19. QUADRATIC FORMULA
This method will work to solve ALL quadratic equations;
however,
for many equations it takes longer than some of the methods
discussed earlier.
The quadratic formula is a good choice if the quadratic
polynomial cannot be factored, the equation cannot be written
as (x+c)
2
= n, or a is not 1 and/or b is an odd number.

20. QUADRATIC FORMULA
Quadratic Formula: x = - (b) ± √b
2
– 4ac
2a
To Solve a Quadratic Using the Quadratic Formula:
Put the quadratic equation into standard form
(above).
Write out the formula and what a, b, & c stand for.
Substitute for each variable.
Split into two separate equations (setting each
equal to zero) and solve.
Check each root in the original equation.

21. QUADRATIC FORMULA (EXAMPLE)
x
2
– 8x – 17 = 0
a = 1
b = –8
c = –17
2
8 ( 8) 4(1)( 17)
2(1)
x
   

8 64 68
2
x
 

8 132
2
x


8 2 33
2
x


4 33

22. 1. m
2
+ 10m + 9 = 0
2. x
2
– 49 = 0
3. z
2
– 4 = 0
4. m
2
– 64 = 0
5. 3x
2
– 12 = 0
6. d
2
– 2d = 0 7 s
2
– s = 0
 

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