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Lesson: Properties of a Discriminant in a Quadratic Equation

Sample Question Videos

Worksheet • 25 Questions • 3 Videos

Q1:

How many non-real roots will a quadratic equation have if its discriminant is positive?

Q2:

How many non-real roots will a quadratic equation have if its discriminant is negative?

Q3:

Determine the type of the roots of the equation 4 π‘₯ ( π‘₯ + 5 ) = βˆ’ 2 5 .

  • Acomplex and not real
  • Breal and equal
  • Creal and different

Q4:

Determine whether the roots of the equation π‘₯ + π‘₯ βˆ’ 2 = 0 2 are rational or not without solving it.

  • Anot rational
  • Brational

Q5:

Which is the correct condition for the quadratic equation π‘Ž π‘₯ + 𝑏 π‘₯ + 𝑐 = 0 2 with real coefficients to have no nonreal roots?

  • A The discriminant 𝑏 βˆ’ 4 π‘Ž 𝑐 2 is nonnegative.
  • BThe discriminant 𝑏 βˆ’ 4 π‘Ž 𝑐 2 is an integer.
  • C The discriminant 𝑏 βˆ’ 4 π‘Ž 𝑐 2 is positive.
  • D The discriminant 𝑏 βˆ’ 4 π‘Ž 𝑐 2 is negative.
  • E The discriminant 𝑏 βˆ’ 4 π‘Ž 𝑐 2 is equal to zero.

Q6:

The roots of the equation 3 π‘₯ βˆ’ ( 4 π‘š βˆ’ 9 ) π‘₯ + π‘š βˆ’ 1 = 0  have different signs. Find the interval in which π‘š lies.

  • A π‘š ∈ ( βˆ’ ∞ , 1 )
  • B π‘š ∈ ( 1 , ∞ )
  • C π‘š ∈ ( βˆ’ ∞ , 1 ]
  • D π‘š ∈ ( βˆ’ ∞ , βˆ’ 1 )
  • E π‘š = 1

Q7:

If the roots of the equation 2 4 π‘₯ + 6 π‘₯ + π‘˜ = 0 2 are not real, find the interval which contains π‘˜ .

  • A π‘˜ ∈  3 8 , ∞ 
  • B π‘˜ ∈  βˆ’ ∞ , 3 8 
  • C π‘˜ ∈  βˆ’ ∞ , 3 8 
  • D π‘˜ ∈  3 8 , ∞ 

Q8:

How many real roots does the equation π‘Ž π‘₯ + 𝑏 π‘₯ + 𝑐 = 0 2 have if π‘Ž β‰  0 and 𝑏 βˆ’ 4 π‘Ž 𝑐 = 0 2 ?

Q9:

Determine the type of the roots of the equation ( π‘₯ βˆ’ 9 ) βˆ’ π‘₯ ( π‘₯ βˆ’ 5 ) = 0 .

  • Acomplex and not real
  • Breal and equal
  • Creal and different

Q10:

Which of the following describes the roots of the equation ?

  • Acomplex and not real
  • Breal and equal
  • Creal and different

Q11:

How many real roots does the following equation have?

  • Atwo roots
  • Bone root
  • Cinfinite number of roots
  • Dno roots

Q12:

If the roots of the equation 2 π‘₯ + 1 0 π‘₯ + 1 2 + 1 π‘˜ = 0 2 are equal, what is the value of π‘˜ ?

Q13:

Are the roots of the equation π‘₯ + 6 π‘˜ π‘₯ + 6 π‘˜ = 1 2 rational for all rational values of π‘˜ ?

  • Ano
  • Byes

Q14:

Given that the roots of the equation βˆ’ 1 8 π‘₯ + 3 π‘˜ π‘₯ βˆ’ 7 2 = 0  are equal, determine all possible values of π‘˜ . For each value of π‘˜ , work out the roots of the equation.

  • A π‘˜ = 2 4 , roots: 2, 2, or π‘˜ = βˆ’ 2 4 , roots: βˆ’ 2 , βˆ’ 2
  • B π‘˜ = βˆ’ 2 4 , roots: 2, 2, or π‘˜ = 2 4 , roots: βˆ’ 2 , βˆ’ 2
  • C π‘˜ = βˆ’ 2 4 , roots: 1 2 , 1 2 , or π‘˜ = 2 4 , roots: βˆ’ 1 2 , βˆ’ 1 2
  • D π‘˜ = 2 4 , roots: 1 2 , 1 2 , or π‘˜ = βˆ’ 2 4 , roots: βˆ’ 1 2 , βˆ’ 1 2

Q15:

Given that π‘š is a real number, and the equation ( 4 π‘š + 8 ) π‘₯ βˆ’ 4 π‘š π‘₯ + π‘š = 0 2 does not have real roots, find the interval which contains π‘š .

  • A ] 0 , ∞ [
  • B ] βˆ’ ∞ , 3 2 ]
  • C ] βˆ’ ∞ , 0 ]
  • D [ 0 , ∞ [
  • E ] βˆ’ ∞ , 0 [

Q16:

If the roots of the equation π‘₯ βˆ’ 8 ( π‘˜ + 1 ) π‘₯ + 6 4 = 0 2 are equal, find the possible values of π‘˜ .

  • A { βˆ’ 3 , 1 }
  • B { βˆ’ 3 3 }
  • C { 3 , βˆ’ 1 }
  • D { 1 , βˆ’ 1 }
  • E { βˆ’ 1 }

Q17:

If the roots of the equation 4 π‘₯ βˆ’ π‘˜ π‘₯ + 1 = 0 2 are equal, what are the possible values of π‘˜ ?

  • A 4 , βˆ’ 4
  • B βˆ’ 4
  • C12
  • D 1 2 , βˆ’ 1 2

Q18:

Given that the equation π‘₯ βˆ’ ( βˆ’ 2 π‘š + 2 8 ) π‘₯ + π‘š = 0 2 2 has no real roots, find the interval that contains π‘š .

  • A π‘š ∈ ] 7 , ∞ [
  • B π‘š ∈ ] βˆ’ ∞ , 7 [
  • C π‘š ∈ ] βˆ’ ∞ , 7 ]
  • D π‘š ∈ [ 7 , ∞ [

Q19:

What type of roots does the equation 6 π‘₯ + π‘˜ π‘₯ + π‘˜ βˆ’ 1 1 = 0 2 have for all real values of π‘˜ ?

  • Areal and equal
  • Breal and different
  • Ccomplex numbers

Q20:

Determine the type of the roots of the equation βˆ’ 2 π‘₯ βˆ’ 6 = 8 π‘₯ + 7 .

  • Areal and different
  • Breal and equal
  • Ccomplex and not real

Q21:

Determine the type of the roots of the equation π‘₯ + 3 6 π‘₯ = 1 2 .

  • Acomplex and not real
  • Breal and equal
  • Creal and different

Q22:

Does the equation π‘₯ + 2 π‘š π‘₯ + π‘š = 9 𝑛 + 8 𝑙 2 2 2 2 have real roots for all real values of π‘š , 𝑛 , and 𝑙 ?

  • Ano
  • Byes

Q23:

Suppose the two roots of the equation π‘₯ βˆ’ ( π‘˜ + 6 ) π‘₯ βˆ’ ( 1 0 π‘˜ βˆ’ 9 ) = 0  are equal. Determine all possible values of π‘˜ , and then find the two roots.

  • A π‘˜ = 0 , two roots: 3 and 3 or π‘˜ = βˆ’ 5 2 , two roots: βˆ’ 2 3 and βˆ’ 2 3
  • B π‘˜ = 0 , two roots: βˆ’ 5 2 and βˆ’ 5 2 or π‘˜ = 3 , two roots: βˆ’ 2 3 and βˆ’ 2 3
  • C π‘˜ = βˆ’ 5 2 , two roots: 3 and 3 or π‘˜ = 0 , two roots: βˆ’ 2 3 and βˆ’ 2 3
  • D π‘˜ = 3 , two roots: βˆ’ 5 2 and βˆ’ 5 2 or π‘˜ = 0 , two roots: βˆ’ 2 3 and βˆ’ 2 3
  • E π‘˜ = 0 , two roots: 3 and 3 or π‘˜ = βˆ’ 2 3 , two roots: βˆ’ 5 2 and βˆ’ 5 2

Q24:

Given that π‘š and 𝑛 are rational, non zero numbers, are the roots of the equation βˆ’ π‘š π‘₯ βˆ’ 3 ο€Ή π‘š βˆ’ 𝑛  π‘₯ + 9 π‘š 𝑛 = 0 2 2 2 2 always rational?

  • Ano
  • Byes

Q25:

If the roots of the equation π‘₯ βˆ’ π‘˜ π‘₯ βˆ’ 4 π‘˜ βˆ’ 4 π‘₯ + 4 = 0 2 are equal, what are the possible values of π‘˜ ? For each value of π‘˜ , work out the roots of the equation.

  • A π‘˜ = 0 , the two roots are 2 and 2 or π‘˜ = βˆ’ 2 4 , the two roots are βˆ’ 1 0 and βˆ’ 1 0
  • B π‘˜ = βˆ’ 2 4 , the two roots are 2 and 2 or π‘˜ = 0 , the two roots are βˆ’ 1 0 and βˆ’ 1 0
  • C π‘˜ = 2 , the two roots are βˆ’ 2 4 and βˆ’ 2 4 or π‘˜ = 0 , the two roots are βˆ’ 1 0 and βˆ’ 1 0
  • D π‘˜ = 0 , the two roots are 2 and 2 or π‘˜ = βˆ’ 1 0 , the two roots are βˆ’ 2 4 and βˆ’ 2 4
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