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Mathematics Dictionary
Dr. K. G. Shih
Inverse of y = a*x^2 + b*x + c


  • Q01 | - Real roots in abs(x^2 - 6*abs(x) + 8) = 0.5
  • Q02 | - Real roots in abs(x^2 - 6*abs(x) + 8) = 1
  • Q03 | - Real roots in abs(x^2 - 6*abs(x) + 8) = 8
  • Q04 | - Real roots in abs(x^2 - 6*abs(x) + 8) = 9
  • Q05 | - Solve abs(x^2 - 6*abs(x) + 8) = d
  • Q06 | - Diagram : y = abs(x^2 - 6*abs(x) + 8) and y = d


Q01. Real roots in abs(x^2 - 6*abs(x) + 8) = 0.5

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
Solution
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Q02. Real roots in abs(x^2 - 6*abs(x) + 8) = 1

It requries to solve 4 equations
  • x^2 - 6*x + 8 = +1
  • x^2 - 6*x + 8 = -1
  • x^2 + 6*x + 8 = +1
  • x^2 + 6*x + 8 = -1
Solutions
  • Equation x^2 - 6*x + 8 = 1
    • x^2 - 6*x + 7 = 0
    • D = b^2 - 4*a*c = 36 - 28 = 8
    • Hence it has two real roots
    • x1 = (6 + Sqr(8))/2 = 4.4142
    • x2 = (6 - Sqr(8))/2 = 1.5858
  • Equation x^2 + 6*x + 8 = 1
    • x^2 + 6*x + 7 = 0
    • D = b^2 - 4*a*c = 36 - 28 = 8
    • Hence it has two real roots
    • x3 = (-6 + Sqr(8))/2 = -1.5858
    • x4 = (-6 - Sqr(8))/2 = -4.4142
  • Equation x^2 - 6*x + 8 = -1
    • x^2 - 6*x + 9 = 0
    • D = b^2 - 4*a*c = 36 - 36 = 0
    • Hence it has two real roots
    • x5 = (6 + Sqr(0))/2 = 3
    • x6 = (6 - Sqr(0))/2 = 3
  • Equation x^2 + 6*x + 8 = -1
    • x^2 + 6*x + 9 = 0
    • D = b^2 - 4*a*c = 36 - 36 = 0
    • Hence it has two real roots
    • x7 = (-6 + Sqr(0))/2 = -3
    • x8 = (-6 - Sqr(0))/2 = -3
Graphic solutions
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Q03 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.
Solutions : It has two different real roots and on real duplicate root
  • Equation x^2 - 6*x + 8 = 8
    • x^2 - 6*x + 0 = 0
    • x1 = 0
    • x2 = 6
  • Equation x^2 + 6*x + 8 = 8
    • x^2 + 6*x + 0 = 0
    • x3 = 0
    • x4 = -6
  • Equation x^2 - 6*x + 8 = -8
    • x^2 - 6*x + 16 = 0
    • D = b^2 - 4*a*c = 36 - 64 = -28
    • Hence it has no real roots
  • Equation x^2 + 6*x + 8 = -8
    • x^2 + 6*x + 16 = 0
    • D = b^2 - 4*a*c = 36 - 64 = -28
    • Hence it has no real roots
Graphic solutions
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Q04 Real roots in abs(x^2 - 6*abs(x) + 8) = 9

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.
Solutions
  • Equation x^2 - 6*x + 8 = 9
    • x^2 - 6*x - 1 = 0
    • D = b^2 - 4*a*c = 36 + 1 = 37
    • Hence it has two real roots
    • x1 = (6 + Sqr(37))/2 =
    • x2 = (6 - Sqr(37))/2 =
  • Equation x^2 + 6*x + 8 = 9
    • x^2 + 6*x - 1 = 0
    • D = b^2 - 4*a*c = 36 + 1 = 37
    • Hence it has two real roots
    • x3 = (-6 + Sqr(37))/2 =
    • x4 = (-6 - Sqr(37))/2 =
  • Equation x^2 - 6*x + 8 = -9
    • x^2 - 6*x + 17 = 0
    • D = b^2 - 4*a*c = 36 - 68 = -32
    • Hence it has no real roots
  • Equation x^2 + 6*x + 8 = -9
    • x^2 + 6*x + 17 = 0
    • D = b^2 - 4*a*c = 36 - 68 = -32
    • Hence it has no real roots
Graphic solutions
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Q05 |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) Hence (1 + d) GT 0 if d GT 0
  • x^2 - 6*x + 8 = -d.
    • x = 3 + sqr(1 - d)
    • x = 3 - sqr(1 - d) Hence (1 - d) GT 0 if d GT 0
  • 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)
Conculsion
  • If all 8 roots are real, then (1 - d) GT 0
  • Since d is positive, hence d is less than 1
Graphic solution
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Q06 Diagram : Number of real roots in abs(x^2 - 6*abs(x) + 8) = d


y = abs(x^2 - 6*x + 8) and y = d


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