Worksheet: Properties of a Discriminant in a Quadratic Equation

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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:

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 𝑥 ( 𝑥 + 5 ) = − 2 5 .

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

Q4:

Determine whether the roots of the equation 𝑥 + 𝑥 − 2 = 0 are rational or not without solving it.

  • Arational
  • Bnot rational

Q5:

Which is the correct condition for the quadratic equation 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 = 0 with real coefficients to have no nonreal roots?

  • AThe discriminant 𝑏 − 4 𝑎 𝑐 is an integer.
  • B The discriminant 𝑏 − 4 𝑎 𝑐 is nonnegative.
  • C The discriminant 𝑏 − 4 𝑎 𝑐 is positive.
  • D The discriminant 𝑏 − 4 𝑎 𝑐 is negative.
  • E The discriminant 𝑏 − 4 𝑎 𝑐 is equal to zero.

Q6:

The roots of the equation 3 𝑥 − ( 4 𝑚 − 9 ) 𝑥 + 𝑚 − 1 = 0 have different signs. Find the interval in which 𝑚 lies.

  • A 𝑚 = 1
  • B 𝑚 ∈ ( − ∞ , 1 ]
  • C 𝑚 ∈ ( − ∞ , 1 )
  • D 𝑚 ∈ ( 1 , ∞ )
  • E 𝑚 ∈ ( − ∞ , − 1 )

Q7:

If the roots of the equation 2 4 𝑥 + 6 𝑥 + 𝑘 = 0 are not real, find the interval which contains 𝑘 .

  • A 𝑘 ∈ − ∞ , 3 8
  • B 𝑘 ∈ 3 8 , ∞
  • C 𝑘 ∈ 3 8 , ∞
  • D 𝑘 ∈ − ∞ , 3 8

Q8:

How many real roots does the equation 𝑎 𝑥 + 𝑏 𝑥 + 𝑐 = 0 have if 𝑎 ≠ 0 and 𝑏 − 4 𝑎 𝑐 = 0 ?

Q9:

Determine the type of the roots of the equation ( 𝑥 − 9 ) − 𝑥 ( 𝑥 − 5 ) = 0 .

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

Q10:

Which of the following describes the roots of the equation − 9 − 1 2 𝑥 = 4 𝑥 ?

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

Q11:

How many real roots does the equation 6 𝑥 + 7 𝑥 − 7 = 0 have?

  • AInfinite number of roots
  • BNo roots
  • COne root
  • DTwo roots

Q12:

If the roots of the equation 2 𝑥 + 1 0 𝑥 + 1 2 + 1 𝑘 = 0 are equal, what is the value of 𝑘 ?

Q13:

Are the roots of the equation 𝑥 + 6 𝑘 𝑥 + 6 𝑘 = 1 rational for all rational values of 𝑘 ?

  • Ayes
  • Bno

Q14:

Given that the roots of the equation − 1 8 𝑥 + 3 𝑘 𝑥 − 7 2 = 0 are equal, determine all possible values of 𝑘 . For each value of 𝑘 , work out the roots of the equation.

  • A 𝑘 = 2 4 , roots: 2, 2, or 𝑘 = − 2 4 , roots: − 2 , − 2
  • B 𝑘 = 2 4 , roots: 1 2 , 1 2 , or 𝑘 = − 2 4 , roots: − 1 2 , − 1 2
  • C 𝑘 = − 2 4 , roots: 1 2 , 1 2 , or 𝑘 = 2 4 , roots: − 1 2 , − 1 2
  • D 𝑘 = − 2 4 , roots: 2, 2, or 𝑘 = 2 4 , roots: − 2 , − 2

Q15:

Given that 𝑚 is a real number, and the equation ( 4 𝑚 + 8 ) 𝑥 − 4 𝑚 𝑥 + 𝑚 = 0 does not have real roots, find the interval which contains 𝑚 .

  • A ( − ∞ , 0 ]
  • B [ 0 , ∞ )
  • C ( 0 , ∞ )
  • D ( − ∞ , 0 )
  • E ( − ∞ , 3 2 ]

Q16:

If the roots of the equation 𝑥 − 8 ( 𝑘 + 1 ) 𝑥 + 6 4 = 0 are equal, find the possible values of 𝑘 .

  • A { 3 , − 1 }
  • B { − 3 , 1 }
  • C { − 3 3 }
  • D { − 1 }
  • E { 1 , − 1 }

Q17:

If the roots of the equation 4 𝑥 − 𝑘 𝑥 + 1 = 0 are equal, what are the possible values of 𝑘 ?

  • A 4 , − 4
  • B12
  • C 1 2 , − 1 2
  • D − 4

Q18:

Given that the equation 𝑥 − ( − 2 𝑚 + 2 8 ) 𝑥 + 𝑚 = 0 has no real roots, find the interval that contains 𝑚 .

  • A 𝑚 ∈ ( 7 , ∞ )
  • B 𝑚 ∈ ( − ∞ , 7 ]
  • C 𝑚 ∈ ( − ∞ , 7 )
  • D 𝑚 ∈ [ 7 , ∞ )

Q19:

What type of roots does the equation 6 𝑥 + 𝑘 𝑥 + 𝑘 − 1 1 = 0 have for all real values of 𝑘 ?

  • Areal and different
  • Breal and equal
  • Ccomplex numbers

Q20:

Determine the type of the roots of the equation − 2 𝑥 − 6 = 8 𝑥 + 7 .

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

Q21:

Determine the type of the roots of the equation 𝑥 + 3 6 𝑥 = 1 2 .

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

Q22:

Does the equation 𝑥 + 2 𝑚 𝑥 + 𝑚 = 9 𝑛 + 8 𝑙 have real roots for all real values of 𝑚 , 𝑛 , and 𝑙 ?

  • Ano
  • Byes

Q23:

Suppose the two roots of the equation 𝑥 − ( 𝑘 + 6 ) 𝑥 − ( 1 0 𝑘 − 9 ) = 0 are equal. Determine all possible values of 𝑘 , and then find the two roots.

  • A 𝑘 = 0 , two roots: − 5 2 and − 5 2 or 𝑘 = 3 , two roots: − 2 3 and − 2 3
  • B 𝑘 = 3 , two roots: − 5 2 and − 5 2 or 𝑘 = 0 , two roots: − 2 3 and − 2 3
  • C 𝑘 = − 5 2 , two roots: 3 and 3 or 𝑘 = 0 , two roots: − 2 3 and − 2 3
  • D 𝑘 = 0 , two roots: 3 and 3 or 𝑘 = − 2 3 , two roots: − 5 2 and − 5 2
  • E 𝑘 = 0 , two roots: 3 and 3 or 𝑘 = − 5 2 , two roots: − 2 3 and − 2 3

Q24:

Given that 𝑚 and 𝑛 are rational, non zero numbers, are the roots of the equation − 𝑚 𝑥 − 3 𝑚 − 𝑛 𝑥 + 9 𝑚 𝑛 = 0 always rational?

  • Ayes
  • Bno

Q25:

If the roots of the equation 𝑥 − 𝑘 𝑥 − 4 𝑘 − 4 𝑥 + 4 = 0 are equal, what are the possible values of 𝑘 ? For each value of 𝑘 , work out the roots of the equation.

  • A 𝑘 = 0 , the two roots are 2 and 2 or 𝑘 = − 1 0 , the two roots are − 2 4 and − 2 4
  • B 𝑘 = 0 , the two roots are 2 and 2 or 𝑘 = − 2 4 , the two roots are − 1 0 and − 1 0
  • C 𝑘 = − 2 4 , the two roots are 2 and 2 or 𝑘 = 0 , the two roots are − 1 0 and − 1 0
  • D 𝑘 = 2 , the two roots are − 2 4 and − 2 4 or 𝑘 = 0 , the two roots are − 1 0 and − 1 0

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