Mathematics Dictionary
Subject : Application of prgram ABH on internat
Keyword : Solve abs(a*x^2 + b*abs(x) + c) = d
Purposes : Use graphs to verify the solution
Subjects
Read Symbol defintion
Q01 |
- How many real roots in abs(x^2 - 6*abs(x) + 8) = 0.5
Q02 |
- How many real roots in abs(x^2 - 6*abs(x) + 8) = 1
Q03 |
- How many real roots in abs(x^2 - 6*abs(x) + 8) = 8
Q04 |
- How many real roots in abs(x^2 - 6*abs(x) + 8) = 9
Q05 |
- |x^2 - 6*|x| + 8 | = d has 8 real roots, find d.
Q06 |
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Q10 |
- Instructions and Tips for program ABH
Q11 |
- Reference
Q1. How many real roots in abs(x^2 - 6*abs(x) + 8) = 0.5
A1. It is equivalent to solve 4 equations :
* Solve x^2 - 6*x + 8 = 0.5
* Solve x^2 + 6*x + 8 = 0.5
* Solve x^2 - 6*x + 8 =-0.5
* Solve x^2 + 6*x + 8 =-0.5
* Using discriminant of quadratic function b^2-4*a*c
* We can proof that it has 8 real roots
* Graphic solution
* Start Softwar ABH
program 07 05
* Give input : 1, -6, 8, 0.5
Go to Begin
Q2. How many real roots in abs(x^2 - 6*abs(x) + 8) = 1
It requries to solve 4 equations
x^2 - 6*x + 8 = +8.
x^2 - 6*x + 8 = -8.
x^2 + 6*x + 8 = +8.
x^2 + 6*x + 8 = -8.
Graphic solutions
Start Softwar ABH
program 07 05
Give input : 1, -6, 8, 1
Go to Begin
Q3 How many real roots in abs(x^2 - 6*abs(x) + 8) = 8
It requries to solve 4 equations
x^2 - 6*x + 8 = +8.
x^2 - 6*x + 8 = -8.
x^2 + 6*x + 8 = +8.
x^2 + 6*x + 8 = -8.
Graphic solutions
Start Softwar ABH
program 07 05
Give input : 1, -6, 8, 8
Go to Begin
Q4. How many real roots in abs(x^2 - 6*abs(x) + 8) = 9
A4. Graphic solution
It requries to solve 4 equations
x^2 - 6*x + 8 = +9.
x^2 - 6*x + 8 = -9.
x^2 + 6*x + 8 = +9.
x^2 + 6*x + 8 = -9.
Graphic solutions
Start Softwar ABH
program 07 05
Give input : 1, -6, 8, 9
Go to Begin
Q5. If |x^2 - 6*|x| + 8| = d has real roots, find d.
It contains the following 4 equations
x^2 - 6*x + 8 = +d.
x^2 - 6*x + 8 = -d.
x^2 + 6*x + 8 = +d.
x^2 + 6*x + 8 = -d.
Using quadratic formula
x^2 - 6*x + 8 = +d.
x = 3 + sqr(1 + d)
x = 3 - sqr(1 + d)
x^2 - 6*x + 8 = -d.
x = 3 + sqr(1 - d)
x = 3 - sqr(1 - d)
x^2 + 6*x + 8 = +d.
x = -3 + sqr(1 + d)
x = -3 - sqr(1 + d)
x^2 + 6*x + 8 = -d.
x = -3 + sqr(1 - d)
x = -3 - sqr(1 - d)
If all 8 roots are real, the 1 - d GT 0
Since d is positive, hence d is less than 1.
Graphic solution
Start Software ABH
program 07 05
Give input : 1, -6, 8, 1
Go to Begin
Go to Begin
Q10 Instructions and Tips for Program ABH
* Q1.
Read Mathematical Symbols
find symbol defintions
* Q2. How to use the program ABH to find answer from graphs ?
1. Click start
2. Click a subject in upper box
3. Click a program in lower box
4. Click Back command and repeat step 3 or steps 2 and 3
* Q3. How to run program 01 03 ?
1. Start the program ABH
2. Click Subject 01 in upper box
3. Click program 03 in lower box
* Q4. How to start the program ?
1. Click
Program ABH
to start
2. Select run at current location (Donot download)
3. Select yes to run
Go to Begin
Q11. Reference
Go to Begin
Go to Begin
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