Algebraic Method to Find the Solution Set of a Quadractic Equation

Algebraic Method ကိုသံုးမယ္ဆိုရင္ ေအာက္ပါေပးထားတဲ့ logical rule ကို သိရပါမယ္။

(i) ab>0 then there are two different possibilities that

(ii) Similarly, for ab<0 the possibilities may be

(i)ab>0 ဆိုသည္မွာ a ႏွင့္ b ေျမႇာက္ျခင္း သည္ အေပါင္းတန္ဘိုး ျဖစ္သည္။ ထိုသို႔ျဖစ္ရန္ a ႏွင့္ b သည္ အေပါင္းတန္ဘိုး ျဖစ္ရမည္။ သို႔မဟုတ္ a ႏွင့္b ႏွစ္ခုလံုးသည္ အႏႈတ္တန္ဘိုး ျဖစ္ရမည္။

(ii) ab<0 ဆိုသည္မွာ a ႏွင့္ b ေျမႇာက္ျခင္း သည္ အႏႈတ္တန္ဘိုး ျဖစ္သည္။ ထိုသို႔ျဖစ္ရန္ a သည္ အေပါင္းတန္ဘိုးဟု ယူဆလွ်င္ b သည္ အႏႈတ္တန္ဘိုး ျဖစ္ရမည္။ သို႔မဟုတ္ a သည္ အႏႈတ္တန္ဘိုးဟု ယူဆလွ်င္ b သည္ အေပါင္းတန္ဘိုး ျဖစ္ရမည္။

Example 1
Use algebraic method, to find the solution set of the inequation 12 - 5x - 2x² ≥ 0 and illustrate it on the number line.

Solution
12 - 5x - 2x² ≥ 0
(4 + x)(3 - 2x) ≥ 0
အထက္တြင္ ေဖၚျပခဲ့ေသာ rule(i)ကို သံုး၍ ေျဖရွင္းပါမည္။

There are two possibilities for (4 + x)(3 - 2x) ≥ 0
(i)(4 + x) ≥ 0 and (3 - 2x) ≥ 0
x ≥ -4 and x ≤ 3/2
ႏွစ္မ်ိဳးလံုးကိုယူရန္မလို၊ အေျခအေန ႏွစ္ရပ္လံုးကို ေျပလည္ေစေသာ အေျဖတစ္ခုကိုသာ ယူမည္။ မည္သည္ကို ယူမည္ ဆိုသည္ကို number line ဆြဲ၍ စဥ္းစားႏိုင္သည္။

အေျခအေန ႏွစ္ရပ္လံုးတြင္ ဘံုျဖစ္ေသာ -4 ≤ x ≤ 3/2ကိုသာ ယူပါမည္။




(ii)(4 + x) ≤ 0 and (3 - 2x) ≤ 0
x ≤ -4 and x ≥ 3/2
အထက္ပါနည္းအတိုင္း number line ဆြဲ၍ စဥ္းစားမည္။

အေျခအေန ႏွစ္ရပ္လံုးတြင္ ေျပလည္ေစေသာ ဘံုအေျဖ မရွိပါ။



The solution set of 12 - 5x - 2x² ≥ 0 is {x/-4 ≤ x ≤ 3/2}.

Example 2

Find the solution set of the inequation 12x² ≥ 10 - 7x by graphical method and illustrate it on the number line.

Solution
12x² ≥ 10 - 7x
12x² + 7x - 10 ≥ 0
(4x + 5)(3x - 2)≥ 0

There are two possibilities for (4 + x)(3 - 2x) ≥ 0
(i)(4x + 5) ≥ 0 and (3x - 2) ≥ 0
Therefore x ≥ - 5/4 and x ≥ 2/3


The solution which satisfies both conditions is
x ≥ 2/3.





(ii)(4x + 5) ≤ 0 and (3x - 2) ≤ 0
Therefore x ≤ - 5/4 and x ≤ 2/3

The solution which satisfies both conditions is
x ≤ - 5/4.

The solution set of 3x² < x² - x + 3 is {x/ x ≤- 5/4 or x ≥ 2/3}.

Graphical Method to Find the Solution Set of a Quadractic Equation

1. ေပးထားေသာ Function ကို y ဟုထားပါ။
2. y=0 ထားၿပီး X-axis ျဖတ္မွတ္မ်ားကို ရွာပါ။
3. x=0 ဟုထားၿပီး Y-axis ျဖတ္မွတ္ကို ရွာပါ။
4. ျဖတ္မွတ္မ်ားကို အသံုးျပဳၿပီး smooth parabola ဆြဲပါ။
5. ေပးေထားေသာ inequation sign ကို ၾကည့္ၿပီး solution set ကို ဆံုးျဖတ္ပါ။

Example 1

Use a graphical method, to find the solution set of the inequation 12 - 5x - 2x² ≥ 0 and illustrate it on the number line.





Solution

Let y=12 - 5x - 2x²

When y = 0,

12 - 5x - 2x² = 0

(4 + x)(3 - 2x) = 0

x = -4 or x = 3/2

Therefore, the graph cuts the X-axis at (-4,0) and (3/2,0).

When x = 0, y = 12

Therefore, the graph cuts the Y-axis at (0,12).



The solution set of 12 - 5x - 2x² ≥ 0 is {x/-4 ≤ x ≤ 3/2}.









Example 2

Find the solution set of the inequation 3x² < x² - x + 3 by graphical method and illustrate it on the number line.



Solution

3x² < x² - x + 3

2x² + x - 3 <0

Let y=2x² + x - 3

When y = 0,

2x² + x - 3 = 0

(2x + 3)(x - 1) = 0

x = - 3/2 or x = 1

Therefore, the graph cuts the X-axis at (-3/2,0) and (1,0).

When x = 0, y = -3

Therefore, the graph cuts the Y-axis at (0,-3).



The solution set of 3x² < x² - x + 3 is {x/-3/2 < x <1}.>







Example 3

Find the solution set of the inequation 12x² ≥ 10 - 7x by graphical method and illustrate it on the number line.



Solution

12x² ≥ 10 - 7x

12x² + 7x - 10 ≥ 0

Let y=12x² + 7x - 10

When y = 0,

12x² + 7x - 10= 0

(4x + 5)(3x - 2) = 0

x = - 5/4 or x = 2/3

Therefore, the graph cuts the X-axis at

(- 5/4,0) and (2/3,0).

When x = 0, y = -10

Therefore, the graph cuts the Y-axis at (0,-10).



The solution set of 3x² < x² - x + 3 is {x/ x ≤- 5/4 or x ≥ 2/3}.

Inequations

ဒီအခန္းမွာေတာ့ Quadratic Inequations ေတြရဲ့ ေပးထားေသာ အေျခအေနကို ေျပလည္ေစမယ့္ solution set ကိုရွာမွာ ျဖစ္ပါတယ္။ Quadratic Inequations ေတြကို မေျဖရွင္းမီ Quadratic Function ဆိုတာကို သိရပါမယ္။

Quadratic Function

An expression f(x) = ax² + bx + c, where a, b, and c are numbers with a≠0 is

called a quadratic function.

ကိန္းရွင္တစ္ခုရဲ့ ႏွစ္ထပ္ကိန္း ပါ၀င္ေသာ function တစ္ခုကို quadratic function လို႔ေခၚပါတယ္။ Quadratic Function ရဲ့ characteristic က x² ပါ။ x² မပါရင္ quadratic function လို႔ မေျပာႏိုင္ပါဘူး။ ဒါေၾကာင့္ coefficient of x² ဟာ 0 မျဖစ္ရပါဘူး။

Quadratic Equation

ax² + bx + c = 0 is called a quadratic equation.

Solutions or Roots of Quadratic Equation





where a≠0









Graph of a Quadratic Function

The graph of a quadratic function is called a parabola. It is basically a curved shape opening up or down.



Quadratic Function တစ္ခုရဲ့ graph ကို parabola လို႔ေခၚပါတယ္။



When you have a quadratic function in the form f(x) = ax² + bx + c

if a > 0, then the parabola opens up ,

coefficient of x² ဟာ positive(a>0) ျဖစ္မယ္ဆိုရင္ graph ဆြဲတဲ့ အခါ open upward parabola ကိုရပါတယ္။

+x² => open upward parabola

if a <0, then the parabola opens down

coefficient of x² ဟာ positive(a<0) ျဖစ္မယ္ဆိုရင္ graph ဆြဲတဲ့ အခါ open downward parabola ကိုရပါတယ္။

- x² =>open downward parabola

Quadratic Inequations

The open sentences ax² + bx + c>0 and ax² + bx + c<0,a ≠ 0 are quadratic inequations in x.

The solution set of the quadratic inequations in x can be found by

(i) Algebraic method

(ii) Graphical method