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+𝑥2=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 𝑏4𝑎𝑐 is positive.
  • BThe discriminant 𝑏4𝑎𝑐 is nonnegative.
  • CThe discriminant 𝑏4𝑎𝑐 is equal to zero.
  • DThe discriminant 𝑏4𝑎𝑐 is negative.
  • EThe discriminant 𝑏4𝑎𝑐 is an integer.

Q6:

The roots of the equation 3𝑥(4𝑚9)𝑥+𝑚1=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.

  • A𝑘38,
  • B𝑘38,
  • C𝑘,38
  • D𝑘,38

Q8:

How many real roots does the equation 𝑎𝑥+𝑏𝑥+𝑐=0 have if 𝑎0 and 𝑏4𝑎𝑐=0?

Q9:

Determine the type of the roots of the equation (𝑥9)𝑥(𝑥5)=0.

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

Q10:

Which of the following describes the roots of the equation 912𝑥=4𝑥?

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

Q11:

How many real roots does the equation 6𝑥+7𝑥7=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+3𝑘𝑥72=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)𝑥4𝑚𝑥+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 𝑥8(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 4𝑥𝑘𝑥+1=0 are equal, what are the possible values of 𝑘?

  • A4,4
  • B4
  • 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+𝑘11=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 2𝑥6=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(10𝑘9)=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 𝑚𝑥3𝑚𝑛𝑥+9𝑚𝑛=0 always rational?

  • ANo
  • BYes

Q25:

If the roots of the equation 𝑥𝑘𝑥4𝑘4𝑥+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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.