Republic of the Philippines
Department of Education
Region III – Central
Luzon Schools Division of
Pampanga
DONNA MARIE D.
TONGOL
sst -III
Write
r
JANE P. VALENCIA ,
EdD
EPS I -
MATHEMATICS
TITLE : Avengers: The Six Infinity Stones
LESSON 5: EQUATIONS TRANSFORMABLE TO QUADRATIC
EQUATIONS
Learning Competency: Solves equation transformable to
quadratic equations (including rational algebraic
equations). M9AL-Ic-d-1
It’s good to meet you, Dr. Banner (Hulk).
I’ ll do anything and Your work on activities that you have done
everything that Mr. Steve and presented was incredible. I’m a huge
fan of the way you lose control and turn
(Captain America)
into an enormous green rage monster.
requires
Congratulations!
to do all the challenges.
Let’s go Iron
Man! We can do
this!
My reflexes are
too fast that’s why
we got the two
infinity stones, we
have 4 stones
Hey Blank Panther! Join me to already!
the next journey to get the
remaining stones by
understanding on how to solve
the equation transforms to
quadratic equation.
The procedures in transforming
these equations to quadratic
equations may also be
different.
Once the equations are
transformed to quadratic
equations, then they can be
solved using the techniques
learned in previous lessons. The
different methods of solving
quadratic equations, such as
extracting square roots,
factoring, completing the square,
and using the quadratic formula,
can be used to solve these
transformed equations.
In solving quadratic equations
that are not written in standard
form we have several examples:
Example 1. Solve x(x– 5) = 36. (This is a quadratic equation
that is not written in standard form.)
Solution: To write the quadratic equation in standard
form, simplify the expression x(x – 5).
x(x – 5) x2 -5x
Rewriting the quadratic equation, x2 -5x -36 =0.
Use any of the four methods of solving quadratic equations in finding
the solutions of the QE.
Factoring is the best way to find the roots of this
equation. x2 -5x -36 =0 (x-9)(x+4)
=0
x-9 =0 or x+4
=0
Then , the roots are x1= 9 and x2 =-4.
Okay, I got it already! Here is
the second example Black
Panther!
Example 2: Find the roots of the equation (x +5)2 + (x -2)2
= 37 .
Solution: The given equation is a quadratic equation but
it is not
written in standard form. Transform this equation to
standard form, then solve it using any of the
methods of solving quadratic equations.
(x +5)2 + (x -2)2 = 37 x2 + 10x + 25 + x2 -4x +4 =37
2 x2 + 6x + 29 = 37
2x2 + 6x + 29 -37 =0
2x2 + 6x - 8 =0, then use
factoring to find the
roots
2x2 + 6x - 8 =0 ( 2x-2 ) (x+4)=0
2x- 2= 0 or x +4
=0
2x = 2 x=
-4 x = 1
The solutions of the equation are: x1 =1 or x2 =- 4.
Hey Iron Man and Black Panther ,
let me help you in solving
rational algebraic equations
transformable to quadratic
equations.
Steps in solving rational algebraic
equations:
1. Find the
LCD.
2. Multiply the equation by the LCD.
3. Simplify and reduce the equation
into quadratic equation.
4. Use any method to solve for the solutions
or roots of
QE.
Here are the examples:
( 4 +𝑥 )
Example
3
1. Solve the equation + = 0.
𝑥 𝑥²
Solution : Determine the LCD and multiply the whole
equation by the LCD
� +𝒙 �
x2 [ + 𝒙 𝒙²
]= 0,
4x + x + 32
=0.
Write equation in standard form , x2 +4x + 3
=0. Solve by factoring or any method .
x2 +4x + 3 =0 ( x+3)
(x+1) =0 x +3 = or x +1 =0
x = -3 , x =-1
The roots are x1 =-3 and x2 =-1.
Did you get it ? Let’s now have the second example:
𝟗 ��
Example 2. Solve for x in the equation𝒙² 𝒙
− + � = �.
Solution: Determine the LCD and multiply the whole equation by
the LCD.
𝟗 ��
x2 [𝒙² − 𝒙 + � = �],
9 -12x +4x2 = 0.
Write the equation in standard form, 4x2 -12x +9
= 0. Solve by quadratic formula or any method .
− ( −12) ± √ (−12) ²−4( 4) ( 9)
x= 2(4)
12 ± √ 144−144
x= 8
12 ± 0
x= 8
,
12+0 3 12−0
x = 8 , x =2 x= 8
, x = .32
� �
The roots are x1 = �
and x2 = .
�
Oh my oh, that was really easy to
solve !
Hey, Mr. Stark ( Iron Man ) and Dr,
Banner( Hulk) you want to have
exciting activities regarding this
equation transformable to quadratic
equation? But, if you can help me, we
can do this
all day.
Alright!
I should not be left in charge of
stuff like this. We will finish the
mission. Let us now answer the
following
activities
.
I will help too! Let’s
get this on!
Transform each of the following equations to quadratic equation in
the form ax2 + bx + c =0. Under each choice is a letter. Write the
letter in the
box corresponding to the number.
1. x (x +5) = 2 x2+5x -2=0 x2+5x =2
x2+5x =2
2. (x + 6)2=15 R A
T
3. (s-4)2 = 12 x2+12x +19=0 x2+12x +36 =15 x2+12x
+51 =0
A N T
s2-8s +28 =0 s2- 8s + 16 =12 s2
4. ( t +2)2 + ( t -3 ) 2
-8s + 4 =0
=0 E F
T
2t2+10t +13=0 2t2-2t +13 =0 2t2- 2t
+36=0
P I
G
1 𝑥
5. +5 x2+8x -16=0 x2 + 6x -4 =0 2t2- 2t
= 𝑥+6 +36=0
𝑥+2
D O G
6𝑥+10 3 6x2+10x +10=0 6x2 -2x+ 3 =0 6x2 +10x
+3 =0
6. 𝑥
+ 𝑥² = 0 H E N
−12+4 𝑥 x2+4x -12=0 x2-2x + 12 =0 x2 +4x
7. +1=
+12=0
0 𝑥² A P
E
( 𝑥 +22 ) ( 𝑥 −2 ) ²
8. 16 + 8x2+8x -112=0 8x2-8x -48 =0 8x2 +8x
= -48=0
5 3 3
F L Y
ANSWER :
1 2 3 4 5 6 7 8
Give the meaning of the word.
Let’s Be True!
Find the solutions of the following equations.
x( x+ 3) =38 x1 =
x2 =
3s(s - 2) = 12 x1 =
x2 =
(t + 1)2 + (t - 8)2 = 45 x1 =
x2 =
(3r + 1)2 + (r + 2)2 = 65 x1 =
x2 =
x1 =
x2 =
Emojitional Equations Activity
Color each square below depending on the color of the correct
answer on each task card. Task cards are numbered. This will guide
you as to what color each square should have depending on the
correct answer.
4 4 4 4 4 4 1 4 1 4 1 4 1 5
5
4 4 4 4 4 4 4 1 4 1 4 1 4 5
5
3 4 4 4 4 3 3 4 1 4 1 4 4 4
5
3 3 4 4 3 3 4 4 4 1 4 4 4 4
4
4 3 3 3 3 4 4 4 4 1 4 4 4 4
4
4 4 3 3 4 4 4 4 3 3 3 3 4 4
4
4 4 3 3 4 4 3 3 3 3 3 3 3 3
4
4 4 3 3 3 3 3 3 3 3 3 4 4 3
3
1 1 3 3 3 3 3 3 3 3 3 4 6 3
3
1 1 3 3 3 3 3 3 3 3 3 3 3 3
3
1 1 1 3 3 3 3 3 3 3 3 3 3 3
3
1 1 1 2 2 2 2 3 3 3 3 3 3 3
3
1 1 1 1 1 2 2 2 2 2 2 2 2 2
1
1 1 1 1 1 1 2 2 2 2 2 2 2 1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 1
1
5. Solve the roots of 1 = A. x= -3 ± √ 5 YELLOW
x+5 . x+2 B. x= 3 ± 2√5 BLACK C. x
x +6 = -3 ± 2√5 GREEN
6. Solve the roots of x-
2 =
4(x
+3)
.
x+3
x –2
A. x1 = -4/3, x2 = 8 RED
B. x1 = 4/3, x2 =-8
GREEN C. x1 = -4/3,
x2 = -8 BLACK
1. Transform x(x-10)= -21 into 3. Transform (x+1)2 +(x-
a quadratic equation in the 2)2=15 into a quadratic
form of ax2 +bx + c=0. equation in the form of ax 2 +bx
+ c=0.
A. x2 –10x –21=0
VIOLET B. x2 +10x + A. 2x2 –4x –25=0 GREEN
21=0 RED B. 2x 4x –5=0
2 – DARK
C. x2—10x +21=0 SKY BLUE BLUE C. 2x 4x +5=0
2 +
ORANGE D.
Transform the equation to quadratic equation, and find the
solutions of each by matching column A with column B.
A Standard Form B
1 ±√13
1. x2 +2x = -1 A. x = 2
2. x2 + 2 = x +5 P. The QE has no real
roots/
3. –(x+2)(x-1)=3 solutions.
E. x = ±√7
1 1 𝑥²
+ =
4. 𝑥−4 𝑥+4 𝑥²−16
N. x =- 1
2 1 I. x = 0 and x =2
− = −1
5. 𝑥+1 𝑥−2
R. x = 1 and x =-3
Excellent Black Panther! It’s my turn, let me do the
last challenge.
Solve . Find the roots of the following quadratic equations.
1 −𝑥 𝑥 −4
+ 1 𝑥²
− =
𝑥 − 4 𝑥 + 4 = 𝑥² − 16 𝑥+3 𝑥−3 𝑥² − 9
𝑥 −3 18
x2 -3 + ( x-3 )2 =2 2 (x-2) 2 - = +
𝑥+4 𝑥 − 2 (𝑥 − 2)(𝑥 + 4)
6=-2
Finally,
we made
it !
ACTIVITY ACTIVITY 2
1
1. x1 = 0, x2 =35
1. R 2. s1 = 4, s2 = 14
2. A 3. t1 = 5, t2 =2
3. 4. r1 = -3, r2 = 2
T 5. x1 = 3, x2 =-2
4.
I
5. O
6. N
7. A
8. L
ACTIVITY 3
4 4 4 4 4 4 1 4 1 4 1 4 1 5 5
4 4 4 4 4 4 4 1 4 1 4 1 4 5 5
3 4 4 4 4 3 3 4 1 4 1 4 4 4 5
3 3 4 4 3 3 4 4 4 1 4 4 4 4 4
4 3 3 3 3 4 4 4 4 1 4 4 4 4 4
4 4 3 3 4 4 4 4 3 3 3 3 4 4 4
4 4 3 3 4 4 3 3 3 3 3 3 3 3 4
4 4 3 3 3 3 3 3 3 3 3 4 4 3 3
1 1 3 3 3 3 3 3 3 3 3 4 6 3 3
1 1 3 3 3 3 3 3 3 3 3 3 3 3 3
1 1 1 3 3 3 3 3 3 3 3 3 3 3 3
1 1 1 2 2 2 2 3 3 3 3 3 3 3 3
1 1 1 1 1 2 2 2 2 2 2 2 2 2 1
1 1 1 1 1 1 2 2 2 2 2 2 2 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
ASSESSMENT CARD ENRICHMENT CARD
1. N
X1 = 0; X=2 x = ± √2
2. A
3. P
4.
I
5. X1 = 2; X1 = -3; X2=2
E X2=1
X1 = 0; X2=4
Mathematics Grade 9 Learner’ Materials First Edition ,pp.90- 100.
Tizon, Lydia and Ulpina , Jisela Naz, JO-ES Publishing House , Math
Builders
2007, Valenzuela City, pp. 113-125.
Solutions to Quadratic Equations with Rational Expressions.
2016.
https://www.analyzemath.com/Algebra2/solutions-to-
equations.html
Solve Quadratic and Rational Equations. 2016.
https://www.analyzemath.com/Algebra2/solve-quadratic-and-
rational- equations.html
