Lesson Worksheet: Discriminants of Quadratics Mathematics

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 4𝑥(𝑥+5)=−25.

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

Q4:

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

  • Anot rational
  • Brational

Q5:

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

  • AThe discriminant 𝑏−4𝑎𝑐 is positive.
  • BThe discriminant 𝑏−4𝑎𝑐 is nonnegative.
  • CThe discriminant 𝑏−4𝑎𝑐 is equal to zero.
  • DThe discriminant 𝑏−4𝑎𝑐 is negative.
  • EThe discriminant 𝑏−4𝑎𝑐 is an integer.

Q6:

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

  • A𝑚=1
  • B𝑚∈(−∞,1]
  • C𝑚∈(−∞,−1)
  • D𝑚∈(1,∞)
  • E𝑚∈(−∞,1)

Q7:

If the roots of the equation 24𝑥+6𝑥+𝑘=0 are not real, find the interval which contains 𝑘.

  • A𝑘∈38,∞
  • B𝑘∈38,∞
  • C𝑘∈−∞,38
  • D𝑘∈−∞,38

Q8:

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

Q9:

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

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

Q10:

Which of the following describes the roots of the equation −9−12𝑥=4𝑥?

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

Q11:

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

  • AInfinite number of roots
  • BOne root
  • CTwo roots
  • DNo roots

Q12:

If the roots of the equation 2𝑥+10𝑥+12+1𝑘=0 are equal, what is the value of 𝑘?

Q13:

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

  • ANo
  • BYes

Q14:

Given that the roots of the equation −18𝑥+3𝑘𝑥−72=0 are equal, determine all possible values of 𝑘. For each value of 𝑘, work out the roots of the equation.

  • A𝑘=−24, roots: 2, 2, or 𝑘=24, roots: −2, −2
  • B𝑘=24, roots: 12, 12, or 𝑘=−24, roots: −12, −12
  • C𝑘=−24, roots: 12, 12, or 𝑘=24, roots: −12, −12
  • D𝑘=24, roots: 2, 2, or 𝑘=−24, roots: −2, −2

Q15:

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

  • A(−∞,0]
  • B(−∞,32]
  • C(−∞,0)
  • D[0,∞)
  • E(0,∞)

Q16:

If the roots of the equation 𝑥−8(𝑘+1)𝑥+64=0 are equal, find the possible values of 𝑘.

  • A{−3,1}
  • B{−1}
  • C{3,−1}
  • D{1,−1}
  • E{−33}

Q17:

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

  • A4,−4
  • B−4
  • C12,−12
  • D12

Q18:

Given that the equation 𝑥−(−2𝑚+28)𝑥+𝑚=0 has no real roots, find the interval that contains 𝑚.

  • A𝑚∈[7,∞)
  • B𝑚∈(−∞,7]
  • C𝑚∈(7,∞)
  • D𝑚∈(−∞,7)

Q19:

What type of roots does the equation 6𝑥+𝑘𝑥+𝑘−11=0 have for all real values of 𝑘?

  • Acomplex numbers
  • Breal and different
  • Creal and equal

Q20:

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

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

Q21:

Determine the type of the roots of the equation 𝑥+36𝑥=12.

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

Q22:

Does the equation 𝑥+2𝑚𝑥+𝑚=9𝑛+8𝑙 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−(10𝑘−9)=0 are equal. Determine all possible values of 𝑘, and then find the two roots.

  • A𝑘=3, two roots: −52 and −52 or 𝑘=0, two roots: −23 and −23
  • B𝑘=0, two roots: 3 and 3 or 𝑘=−52, two roots: −23 and −23
  • C𝑘=−52, two roots: 3 and 3 or 𝑘=0, two roots: −23 and −23
  • D𝑘=0, two roots: −52 and −52 or 𝑘=3, two roots: −23 and −23
  • E𝑘=0, two roots: 3 and 3 or 𝑘=−23, two roots: −52 and −52

Q24:

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

  • ANo
  • BYes

Q25:

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

  • A𝑘=2, the two roots are −24 and −24 or 𝑘=0, the two roots are −10 and −10
  • B𝑘=0, the two roots are 2 and 2 or 𝑘=−24, the two roots are −10 and −10
  • C𝑘=−24, the two roots are 2 and 2 or 𝑘=0, the two roots are −10 and −10
  • D𝑘=0, the two roots are 2 and 2 or 𝑘=−10, the two roots are −24 and −24

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