Mathematics Grade 9

2. • Dear God,
• we humbly gather together virtually.
• Thinking of starting today's online class.
• Oh, father, we come to you asking for your
guidance, wisdom, and support as we wait
to be pumped with knowledge by our
teachers.
• Please help us understand whatever we
are being taught, so we can have a
brighter future when we finish our
program.
• We pray this trusting your holy name.
• Amen.

3. REMINDERS
2.Always turn on your
camera.
1.Be on time.
3.Unmute your audio
when called by your
teacher.
4.Do have your
module, notebook and
pen.
6.Be attentive and
enjoy learning.
5.Questions will be
entertained after a
discussed topic.

4. ATTENDANCE

5. WEEKLY TASK
SEPT. 3, 2022 (Saturday)
►Answer the Pretest Item #
1-5 on pp. 1-2 of your
Module

6. MATH 9- QUARTER 1 -
WEEK 1
QUADRATIC
EQUATIONS

7. PIC
-a-
QUAD

8. Extracting Square Roots

9. Factoring

10. Completing the Square

11. Using the Quadratic

12. REVIEW:
DEFINITIONS: A quadratic equation in one variable is a
mathematical sentence of degree 2 that can be written in the
standard form ax2 + bx + c = 0, where 𝑎,𝑏, and 𝑐 are real numbers
and a ≠ 0. The values of 𝑎,𝑏, and 𝑐 are the numerical coefficients
of quadratic term, linear term, and constant term, respectively.
ax2 + bx + c = 0
Constant term
Linear term
Quadratic
term

13. LESSON 2: SOLVING QUADRATIC
EQUATIONS
The solution set or roots
of quadratic equations
are values of the
variable that will satisfy
the equation.

14. METHODS USED TO
SOLVE QUADRATIC EQUATIONS
1. Extracting Square Roots
2. Factoring
3. Completing the Square
4. Quadratic Formula

15. 1. EXTRACTING SQUARE
ROOTS/SQUARE ROOT
PROPERTY
•

16. Example 1
x2 – 100 = 0
x = ± 10
3x2 + 15 = 0
x2 – 100 + 100 = 0 + 100
x2 = 100
Solution Set:
{10, -10}
Example 5
Example 4
Example 3
Example 2
x2 = 0
x = 0
Solution Set:
{0}
3x2 + 15 (-15) = 0 + (-15)
3x2 = -15
x2 = -5
Solution Set:
No Solution
2x2 - 6 = 0
2x2 = 6
x2 = 3
(x – 5)2 = 16
x - 5 = ±4
x – 5 + 5=±4 +5
x = 5 ± 4
Solution Set:
{9, 1}
x = 5 + 4
x = 9 x = 1
x = 5 - 4

17. 2. FACTORING
Factoring is typically one of the easiest
and quickest ways to solve quadratic
equations;
however,
Not all quadratic polynomials can be
factored. Applicable whenever the
polynomial in the equation is factorable.
This means that factoring will not work
to solve many quadratic equations.

18. EXAMPLE 1:
STEPS
Solve: 𝟑𝒙 2 − 𝟏𝟓𝒙 = 𝟎
𝟑𝒙(𝒙 – 5) = 𝟎
𝟑𝒙 = 0
𝒙 = 0
𝒙 – 5 = 0
𝒙 = 5
or
The roots are 0
and 5.
Transform the equation in
standard form if necessary
Factor the Polynomial
Apply the Zero Product
Property by setting each factor
equal to zero
Solve the equation
𝟑 𝟑

19. EXAMPLE 2: STEPS
Solve: 𝒙 2 = 9𝒙 - 18
𝒙 2 – 9x + 18 = 9x – 9x -18 + 18
𝒙 - 6 = 0
𝒙 = 6
𝒙 – 3 = 0
𝒙 = 3
or
The roots are 6 and
Transform the equation in
standard form if necessary
Write the equation in Standard
Form by adding -9x and 18 on
both sides.
Apply the Zero Product
Property by setting each factor
equal to zero
Solve the equation
Simplify by combining like
terms.
𝑥 2 − 9𝑥 + 18 = 0
(𝑥 – 6) (𝑥 – 3) = 0 Factor the Polynomial.

20. 3. 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.

21. 3. COMPLETING THE SQUARE
Before we use this method for solving quadratic equation, let us recall that
if the expression 𝑎𝑥2 + 𝑏𝑥 + 𝑐 is a perfect square trinomial, it can be factored
out into two identical binomial factors.
𝑥 2 + 4𝑥 + 4 𝑥 2 - 10𝑥 + 25 4𝑥 2 + 12𝑥 + 9
= (𝑥 + 2)(𝑥 +
2)
= (𝑥 + 2) 2 = (2𝑥 + 3) 2
= (𝑥 - 5) 2
= (𝑥 - 5)(𝑥 - 5) = (2𝑥 + 3)(2𝑥 + 3)

22. EXAMPLE 1 X2 + 12X = 0
• x2 + 12x = 0
•
• x2 + 12x + ___ = 0 + ___
• b= 12 12 / 2 =6
• 62=36
• x2 + 12x + 36 = 0 + 36
• (x + 6) 2 = 36
• √(x + 6) 2 = √36
• x + 6 = +6
• x + 6 = +6 x + 6 = -6
• -6 = -6 - 6 = -6
• x = 0 x = -12
• The roots are 0 and -12.
• 1. Make one side a perfect square. Move
the quadratic term and linear term to the
left side of the equation
• 2. Add a blank to both sides
• 3. Divide “b” by 2
• 4. Square that answer.
• 5. Add it to both sides
• 6. Factor 1st side
• 7. Square root both sides
• 8. Solve for x

23. EXAMPLE 2 X2 + 6X + 8 = 0
• x2 + 6x + 8 = 0
• x2 + 6x + 8 -8 = 0 - 8
• x2 + 6x + ___ = -8 + ___
• b= 6 6 / 2 =3
• 32=9
• x2 + 6x + 9 = -8 + 9
• (x + 3) 2 = 1
• √(x + 3) 2 = √1
• x + 3 = +1
• x + 3 = +1 x + 6 = -1
• - 3 = -3 - 3 = -3
• x = -2 x = -4
• The roots are -2 and -4.
• 1. Move constant to other side. Move the
quadratic term and linear term to the left
side of the equation
• 2. Add a blank to both sides
• 3. Divide “b” by 2
• 4. Square that answer.
• 5. Add it to both sides
• 6. Factor 1st side
• 7. Square root both sides
• 8. Solve for x

24. 4. 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.

25. Solve using the Quadratic Formula.
1.) x2 = x + 20
1x2 + (–1x) + (–20) = 0
a=1 b=-1 c=-20
Write in standard form. Identify
a, b, and c.
Use the quadratic formula.
Simplify.
Substitute 1 for a, –1 for b, and
–20 for c.

26. Solve using the Quadratic Formula.
x = 5 or x = –4
Simplify.
Write as two equations.
Solve each equation.
x2 = x + 20

27. Solve using the Quadratic Formula.
2.) –3x2 + 5x + 2 = 0
Identify a, b, and c.
Use the Quadratic Formula.
Substitute –3 for a, 5 for b,
and 2 for c.
Simplify.
–3x2 + 5x + 2 = 0
a=-3 b=5 c=2

28. Solve using the Quadratic Formula.
Simplify.
Write as two equations.
Solve each equation.
x = – or x = 2
–3x2 + 5x + 2 = 0

29. Solve using the Quadratic Formula.
3.) 2 – 5x2 = –9x
Write in standard form. Identify
a, b, and c.
(–5)x2 + 9x + (2) = 0
a=-5 b=9 c=2
Use the Quadratic Formula.
Substitute –5 for a, 9 for b,
and 2 for c.
Simplify.

30. Method Most appropriate to use
Extracting Square Roots or Square- Root
Property
for quadratic equations of the form 𝑥2 =
𝑘
Factoring If the constant term is 0, or if 𝑎𝑥2 + 𝑏𝑥 + 𝑐
is factorable.
Completing the square For any quadratic equation of the form
𝑎𝑥2 + 𝑏𝑥 + 𝑐, where a is 1 and b is even.
Quadratic formula For any quadratic equation.
REMEMBER

31. EXERCISES: SOLVE THE FOLLOWING USING
THE APPROPRIATE METHOD OF SOLVING
QUADRATIC EQUATIONS.
1.
2.
3.
4.
5.
2x2 – 2x + 3 = 0
x2 + 8x - 84 = 0
x2 = 121
x2 - 4x - 5 = 0
2x2 = – 5x + 10