Worksheet: Discriminants of Quadratics

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 4π‘₯(π‘₯+5)=βˆ’25.

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

Q4:

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

  • Anot rational
  • Brational

Q5:

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

  • AThe discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is positive.
  • BThe discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is nonnegative.
  • CThe discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is equal to zero.
  • DThe discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is negative.
  • EThe discriminant π‘βˆ’4π‘Žπ‘οŠ¨ is an integer.

Q6:

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

  • Aπ‘š=1
  • Bπ‘šβˆˆ(βˆ’βˆž,1]
  • Cπ‘šβˆˆ(βˆ’βˆž,βˆ’1)
  • Dπ‘šβˆˆ(1,∞)
  • Eπ‘šβˆˆ(βˆ’βˆž,1)

Q7:

If the roots of the equation 24π‘₯+6π‘₯+π‘˜=0 are not real, find the interval which contains π‘˜.

  • Aπ‘˜βˆˆο€Ό38,∞
  • Bπ‘˜βˆˆο”38,∞
  • Cπ‘˜βˆˆο€Όβˆ’βˆž,38
  • Dπ‘˜βˆˆο€Όβˆ’βˆž,38

Q8:

How many real roots does the equation π‘Žπ‘₯+𝑏π‘₯+𝑐=0 have if π‘Žβ‰ 0 and π‘βˆ’4π‘Žπ‘=0?

Q9:

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

Q10:

Which of the following describes the roots of the equation βˆ’9βˆ’12π‘₯=4π‘₯?

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

Q11:

How many real roots does the equation 6π‘₯+7π‘₯βˆ’7=0 have?

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

Q12:

If the roots of the equation 2π‘₯+10π‘₯+12+1π‘˜=0 are equal, what is the value of π‘˜?

Q13:

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

  • Ano
  • Byes

Q14:

Given that the roots of the equation βˆ’18π‘₯+3π‘˜π‘₯βˆ’72=0 are equal, determine all possible values of π‘˜. For each value of π‘˜, work out the roots of the equation.

  • Aπ‘˜=βˆ’24, roots: 2, 2, or π‘˜=24, roots: βˆ’2, βˆ’2
  • Bπ‘˜=24, roots: 12, 12, or π‘˜=βˆ’24, roots: βˆ’12, βˆ’12
  • Cπ‘˜=βˆ’24, roots: 12, 12, or π‘˜=24, roots: βˆ’12, βˆ’12
  • Dπ‘˜=24, roots: 2, 2, or π‘˜=βˆ’24, roots: βˆ’2, βˆ’2

Q15:

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

  • A(βˆ’βˆž,0]
  • B(βˆ’βˆž,32]
  • C(βˆ’βˆž,0)
  • D[0,∞)
  • E(0,∞)

Q16:

If the roots of the equation π‘₯βˆ’8(π‘˜+1)π‘₯+64=0 are equal, find the possible values of π‘˜.

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

Q17:

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

  • A4,βˆ’4
  • Bβˆ’4
  • C12,βˆ’12
  • D12

Q18:

Given that the equation π‘₯βˆ’(βˆ’2π‘š+28)π‘₯+π‘š=0 has no real roots, find the interval that contains π‘š.

  • Aπ‘šβˆˆ[7,∞)
  • Bπ‘šβˆˆ(βˆ’βˆž,7]
  • Cπ‘šβˆˆ(7,∞)
  • Dπ‘šβˆˆ(βˆ’βˆž,7)

Q19:

What type of roots does the equation 6π‘₯+π‘˜π‘₯+π‘˜βˆ’11=0 have for all real values of π‘˜?

  • Acomplex numbers
  • Breal and different
  • Creal and equal

Q20:

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

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

Q21:

Determine the type of the roots of the equation π‘₯+36π‘₯=12.

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

Q22:

Does the equation π‘₯+2π‘šπ‘₯+π‘š=9𝑛+8π‘™οŠ¨οŠ¨οŠ¨οŠ¨ 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’(10π‘˜βˆ’9)=0 are equal. Determine all possible values of π‘˜, and then find the two roots.

  • Aπ‘˜=3, two roots: βˆ’52 and βˆ’52 or π‘˜=0, two roots: βˆ’23 and βˆ’23
  • Bπ‘˜=0, two roots: 3 and 3 or π‘˜=βˆ’52, two roots: βˆ’23 and βˆ’23
  • Cπ‘˜=βˆ’52, two roots: 3 and 3 or π‘˜=0, two roots: βˆ’23 and βˆ’23
  • Dπ‘˜=0, two roots: βˆ’52 and βˆ’52 or π‘˜=3, two roots: βˆ’23 and βˆ’23
  • Eπ‘˜=0, two roots: 3 and 3 or π‘˜=βˆ’23, two roots: βˆ’52 and βˆ’52

Q24:

Given that π‘š and 𝑛 are rational, non zero numbers, are the roots of the equation βˆ’π‘šπ‘₯βˆ’3ο€Ήπ‘šβˆ’π‘›ο…π‘₯+9π‘šπ‘›=0 always rational?

  • Ano
  • Byes

Q25:

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

  • Aπ‘˜=2, the two roots are βˆ’24 and βˆ’24 or π‘˜=0, the two roots are βˆ’10 and βˆ’10
  • Bπ‘˜=0, the two roots are 2 and 2 or π‘˜=βˆ’24, the two roots are βˆ’10 and βˆ’10
  • Cπ‘˜=βˆ’24, the two roots are 2 and 2 or π‘˜=0, the two roots are βˆ’10 and βˆ’10
  • Dπ‘˜=0, the two roots are 2 and 2 or π‘˜=βˆ’10, the two roots are βˆ’24 and βˆ’24

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