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Worksheet: Properties of a Discriminant in a Quadratic Equation

In this worksheet, we will practice finding the discriminant of a quadratic equation and using it to determine the number and type of its roots (solution) without solving it.

Q1:

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

  • Areal and different
  • Bcomplex and not real
  • Creal and equal

Q2:

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

Q3:

Given that the roots of the equation 4 π‘₯ βˆ’ 1 2 π‘₯ + π‘˜ = 0 2 are real and different, find the interval which contains π‘˜ .

  • A π‘˜ ∈ [ 9 , ∞ )
  • B π‘˜ = 9
  • C π‘˜ ∈ ( βˆ’ ∞ , 9 ]
  • D π‘˜ ∈ ( βˆ’ ∞ , 9 )
  • E π‘˜ ∈ ( 9 , ∞ )

Q4:

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 ?

  • Ayes
  • Bno

Q5:

How many real roots does the following equation have?

  • Ano roots
  • Bone root
  • Cinfinite number of roots
  • Dtwo roots

Q6:

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

Q7:

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

Q8:

Which of the following describes the roots of the equation ?

  • Areal and different
  • Bcomplex and not real
  • Creal and equal

Q9:

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‘˜ ?

Q10:

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

  • Arational
  • Bnot rational

Q11:

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?

  • Ayes
  • Bno

Q12:

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’ 1 0 , the two roots are βˆ’ 2 4 and βˆ’ 2 4
  • 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’ 2 4 , the two roots are βˆ’ 1 0 and βˆ’ 1 0

Q13:

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

  • Ayes
  • Bno

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: 1 2 , 1 2 , or π‘˜ = βˆ’ 2 4 , roots: βˆ’ 1 2 , βˆ’ 1 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: 2, 2, or π‘˜ = βˆ’ 2 4 , roots: βˆ’ 2 , βˆ’ 2

Q15:

Given that is a real number, and the equation does not have real roots, find the interval which contains .

  • A
  • B
  • C
  • D
  • E

Q16:

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

  • Areal and different
  • Bcomplex and not real
  • Creal and equal

Q17:

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

Q18:

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

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

Q19:

For what value of 𝐾 does the equation 3 𝑦 = 5 𝑦 βˆ’ 𝐾 2 have exactly one solution?

  • A βˆ’ 5
  • B βˆ’ 2 5 1 2
  • C10
  • D 2 5 1 2
  • E 5 3

Q20:

Given that the equation has no real roots, find the interval that contains .

  • A
  • B
  • C
  • D

Q21:

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

  • Acomplex numbers
  • Breal and equal
  • Creal and different

Q22:

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

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

Q23:

Determine whether the roots of the equation π‘₯ βˆ’ √ 5 π‘₯ βˆ’ 1 = 0 2 are rational or not without solving it.

  • Anot rational
  • Brational

Q24:

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

  • Areal and different
  • Bcomplex and not real
  • Creal and equal

Q25:

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

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