The document outlines methods for solving quadratic equations, including factoring and using the quadratic formula. It emphasizes steps such as rewriting quadratic equations in standard form, applying the zero product property, and solving resulting linear equations. Additionally, it covers special cases like perfect square trinomials and provides examples for clarity.

1. Grade 9 – Mathematics
Quarter I
SOLVING QUADRATIC
EQUATIONS BY FACTORING
& QUADRATIC FORMULA

2. OBJECTIVES:
•solve quadratic equations by factoring in the
form 𝑎𝑥2 + bx = 0; and 𝑎𝑥2 + bx + c = 0.
•solve quadratic equations using the
quadratic formula.

3. FACTORING may also be used to solve a quadratic equation when
none of the constants 𝑎, 𝑏, 𝑜𝑟 𝑐 is 0.
ZERO PRODUCT PROPERTY
If 𝑎𝑏 = 0, then either 𝑎 = 0 or 𝑏 = 0, or both a and b are 0.
Factoring Method
1.Write the equation in the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0.
2.Factor the left-hand side of the equation.
3.Set each factor equal to zero using the Principle of Zero
Products.
4.Solve each resulting linear equation.

4. Solve. 9𝑥2 − 4 = 0
Factor. 3𝑥 + 2 3𝑥 − 2 = 0
Set each factor to 0. 3x + 2 = 0
3𝑥 = −2
2
𝑥 = −
3
3x − 2 = 0
3x = 2
2
𝑥 =
3
FACTORING THE
DIFFERENCE OF
TWO SQUARES
TAKE NOTE:
(a + b) and (a – b) are sum and
difference of two terms
respectively. Their product
a2 − b2 is a difference of two
squares

5. Solve.2𝑥2 − 5𝑥 − 3 = 0
Factor.
Set each factor to 0. x − 3 = 0
x = 3
2x + 1 = 0
2𝑥 = −1
1
𝑥 = −
2
− 3
1 x
2x = 0
+
FACTORING
GENERAL
TRINOMIAL

6. 5 2y
Solve. 4y2+ 20y =−25
Transform into 𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎.
Factor. 5
2y
4y 2+20y+25 =0
Set each factor to 0. 2y − 5=0
y = − 5
2
FACTORING
PERFECT SQUARE
TRINOMIAL
− −
−
=0

7. Note: When constant is 0, the quadratic equation will be
of the form 𝒂𝒙𝟐 + 𝒃𝒙 = 𝟎.
Solve for 3𝑥2 + 18𝑥 = 0
Factor.
Set each factor to 0.
Solve for x.
3𝑥 𝑥 + 6 = 0
3𝑥 = 0 x + 6 = 0
𝑥 = 0 x = −6
For equations of this forms, one root will
always be equal to zero.
3, 18 = 3
𝑥, 𝑥2 = 𝑥

8. Solve for 4𝑥2 − 2𝑥 = 0
Factor. 2𝑥 2𝑥 − 1 = 0
Set each factor to 0.
Solve for x.
2𝑥 = 0
𝑥 = 0
2x − 1 = 0
2x = 1
4, 2 = 2
𝑥, 𝑥2 = 𝑥
1
x =
2
The roots of quadratic equation of the form
𝒂
𝑎𝑥2 + 𝑏𝑥 = 0 are 𝒙 = 𝟎 and 𝒙 = −𝒃
.

9. 𝑆𝑜𝑙𝑣𝑒. 7𝑥2 + 18𝑥 = 10𝑥2 + 12𝑥
7𝑥2 − 10𝑥2 + 18𝑥 − 12𝑥 = 0
−3𝑥2 + 6𝑥 = 0
𝒂
𝒙 = 𝟎 and 𝒙 = −𝒃
. 𝑎
−𝑏 −6
𝑥 = = = 2
−3
𝒙 = 𝟎 𝒙 = 𝟐

10. Solve the equations without factoring, Instead, use the fact that
𝒂
the roots of 𝑎𝑥2 + 𝑏𝑥 = 0 are 𝒙 = 𝟎 and 𝒙 = −𝒃
.
a. 𝑥2 + 3𝑥 = 0
b. 2𝑥2 + 8𝑥 = 0
c. 9𝑥2 − 𝑥 = 0
d. 4𝑥2 − 10𝑥 = 0
thus, 𝑥 = 0 and 𝑥 = −3.
thus, 𝑥 = 0 and 𝑥 = −4.
9
thus, 𝑥 = 0 and 𝑥 = 1
.
thus, 𝑥 = 0 and 𝑥 =
5
2
.

11. QUADRATIC FORMULA
𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
• Write the quadratic equation in general form
• Substitute the value of a, b, and c in the quadratic
formula
• Solve the quadratic equation

12. Solve: 7𝑥2+18𝑥= − 10
𝑥 =
−𝑏 ± 𝑏2 − 4𝑎𝑐
2𝑎
a = 7; b = 18; c = 10
7𝑥2+18𝑥+10 = 0
𝑥 =
−18 ± 182 − 4(7)(10)
2(7)
𝑥 =
−18 ± 324 − 280
14
𝑥 =
−18 ± 44
14
𝑥 =
−18 ± 4 𝑥 11
14
𝑥 =
−18 ± 2 11
14
𝑥 =
−18 − 2 11
14
𝑥 =
−18 + 2 11
14
or
1st
2nd
3rd 4th
5th 6th 7th
8th