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Lesson: Writing a Quadratic Equation given the Roots

Sample Question Videos

Worksheet • 25 Questions • 3 Videos

Q1:

Given that 𝐿 + 3 and 𝑀 + 3 are the roots of the equation π‘₯ + 8 π‘₯ + 1 2 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 and 𝑀 .

  • A π‘₯ + 1 4 π‘₯ + 4 5 = 0 2
  • B π‘₯ + 1 7 π‘₯ + 3 4 = 0 2
  • C π‘₯ βˆ’ 1 7 π‘₯ + 3 1 = 0 2
  • D π‘₯ + 1 7 π‘₯ + 1 9 = 0 2
  • E π‘₯ βˆ’ 1 7 π‘₯ + 1 9 = 0 2

Q2:

If 𝐿 and 𝑀 are the roots of the equation π‘₯ βˆ’ 1 9 π‘₯ + 9 = 0  , find, in its simplest form, the quadratic equation whose roots are 𝐿 βˆ’ 2 and 𝑀 βˆ’ 2 .

  • A π‘₯ βˆ’ 1 5 π‘₯ βˆ’ 2 5 = 0 
  • B π‘₯ βˆ’ 2 3 π‘₯ + 3 2 = 0 
  • C π‘₯ + 1 5 π‘₯ βˆ’ 2 5 = 0 
  • D π‘₯ βˆ’ 1 5 π‘₯ + 3 2 = 0 
  • E π‘₯ βˆ’ 2 3 π‘₯ βˆ’ 2 5 = 0 

Q3:

Given that 𝐿 and 𝑀 are the roots of the equation π‘₯ + π‘₯ βˆ’ 2 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 + 𝑀 2 and 𝑀 + 𝐿 2 .

  • A π‘₯ βˆ’ 4 π‘₯ βˆ’ 5 = 0 2
  • B π‘₯ βˆ’ 4 π‘₯ + 9 = 0 2
  • C π‘₯ + 4 π‘₯ βˆ’ 5 = 0 2
  • D π‘₯ βˆ’ π‘₯ βˆ’ 5 = 0 2
  • E π‘₯ + π‘₯ βˆ’ 5 = 0 2

Q4:

Given that 𝐿 and 𝑀 are the roots of the equation 3 π‘₯ + 1 6 π‘₯ βˆ’ 1 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 2 and 𝑀 2 .

  • A 1 2 π‘₯ + 3 2 π‘₯ βˆ’ 1 = 0 2
  • B π‘₯ βˆ’ 3 2 π‘₯ βˆ’ 1 = 0 2
  • C 1 2 π‘₯ βˆ’ 3 2 π‘₯ βˆ’ 1 = 0 2
  • D π‘₯ + 3 2 π‘₯ βˆ’ 1 = 0 2
  • E 1 2 π‘₯ + 3 2 π‘₯ + 1 = 0 2

Q5:

If 𝐿 and 𝑀 are the roots of the equation 2 π‘₯ βˆ’ 3 π‘₯ + 1 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 2 𝐿 2 and 2 𝑀 2 .

  • A 2 π‘₯ βˆ’ 5 π‘₯ + 2 = 0 2
  • B π‘₯ βˆ’ 5 π‘₯ + 1 = 0 2
  • C 2 π‘₯ βˆ’ 5 π‘₯ + 1 = 0 2
  • D π‘₯ βˆ’ 5 π‘₯ + 2 = 0 2
  • E 2 π‘₯ + 5 π‘₯ + 2 = 0 2

Q6:

Given that 𝐿 and 𝑀 are the roots of the equation 3 π‘₯ + 6 π‘₯ + 2 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 𝑀 2 and 𝑀 𝐿 2 .

  • A 3 π‘₯ + 1 8 π‘₯ + 2 = 0 2
  • B 3 π‘₯ + 6 π‘₯ + 2 = 0 2
  • C 3 π‘₯ βˆ’ 1 8 π‘₯ + 2 = 0 2
  • D π‘₯ βˆ’ 1 8 π‘₯ + 2 = 0 2
  • E 3 π‘₯ βˆ’ 1 8 π‘₯ βˆ’ 2 = 0 2

Q7:

Given that 𝐿 and 𝑀 are the roots of the equation π‘₯ + 3 π‘₯ βˆ’ 5 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 𝑀 2 and 𝑀 𝐿 2 .

  • A π‘₯ βˆ’ 1 5 π‘₯ βˆ’ 1 2 5 = 0 2
  • B π‘₯ + 8 π‘₯ + 2 5 = 0 2
  • C π‘₯ + 1 5 π‘₯ βˆ’ 1 2 5 = 0 2
  • D π‘₯ βˆ’ 8 π‘₯ βˆ’ 1 2 5 = 0 2
  • E π‘₯ + 1 5 π‘₯ + 2 5 = 0 2

Q8:

The roots of the equation π‘₯ + 6 π‘₯ + 𝑐 = 0 2 are 𝐿 and 𝑀 , where 𝐿 + 𝑀 = 2 6 2 2 . Find the value of 𝑐 , and determine, in its simplest form, the equation whose roots are 𝐿 𝑀 + 𝑀 𝐿 2 2 and 𝐿 𝑀 .

  • A 𝑐 = 5 , π‘₯ + 2 5 π‘₯ βˆ’ 1 5 0 = 0 2
  • B 𝑐 = 3 1 , π‘₯ βˆ’ 2 5 π‘₯ + 3 5 = 0 2
  • C 𝑐 = βˆ’ 5 , π‘₯ βˆ’ 2 5 π‘₯ + 1 5 0 = 0 2
  • D 𝑐 = βˆ’ 1 0 , π‘₯ + 3 5 π‘₯ + 1 5 0 = 0 2
  • E 𝑐 = 1 0 , π‘₯ βˆ’ 2 5 π‘₯ + 1 5 0 = 0 2

Q9:

Given that 𝐿 and 𝑀 are the roots of the equation π‘₯ βˆ’ 3 π‘₯ + 1 2 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 1 𝐿 2 and 1 𝑀 2 .

  • A 1 4 4 π‘₯ + 1 5 π‘₯ + 1 = 0 2
  • B 1 4 4 π‘₯ + 5 π‘₯ + 1 = 0 2
  • C 1 4 4 π‘₯ βˆ’ 1 5 π‘₯ + 1 = 0 2
  • D π‘₯ βˆ’ 1 5 π‘₯ + 1 = 0 2
  • E 1 4 4 π‘₯ βˆ’ 1 5 π‘₯ βˆ’ 1 = 0 2

Q10:

Given that 𝐿 and 𝑀 are the roots of the equation π‘₯ βˆ’ 1 3 π‘₯ βˆ’ 5 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 + 1 and 𝑀 + 1 .

  • A π‘₯ βˆ’ 1 5 π‘₯ + 9 = 0 2
  • B π‘₯ + 1 1 π‘₯ + 8 = 0 2
  • C π‘₯ βˆ’ 1 1 π‘₯ + 9 = 0 2
  • D π‘₯ βˆ’ 1 5 π‘₯ + 8 = 0 2
  • E π‘₯ + 1 5 π‘₯ + 9 = 0 2

Q11:

Given that 1 𝑀 and 1 𝐿 are the roots of the equation π‘₯ βˆ’ 8 π‘₯ βˆ’ 1 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 𝑀 + 3 and 𝐿 + 𝑀 + 6 .

  • A π‘₯ βˆ’ 4 = 0 2
  • B π‘₯ + 9 π‘₯ + 3 6 = 0 2
  • C π‘₯ + 4 π‘₯ βˆ’ 4 = 0 2
  • D π‘₯ + 4 = 0 2
  • E π‘₯ βˆ’ 4 π‘₯ βˆ’ 4 = 0 2

Q12:

Given that 𝐿 and 𝑀 are the roots of the equation π‘₯ βˆ’ 9 π‘₯ βˆ’ 7 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 βˆ’ 𝑀 and 𝑀 βˆ’ 𝐿 .

  • A π‘₯ βˆ’ 1 0 9 = 0 2
  • B π‘₯ βˆ’ 1 8 π‘₯ βˆ’ 1 0 9 = 0 2
  • C π‘₯ + 1 0 9 = 0 2
  • D π‘₯ + 1 8 π‘₯ βˆ’ 1 0 9 = 0 2
  • E π‘₯ βˆ’ 9 5 = 0 2

Q13:

If 𝐿 and 𝑀 are the roots of the equation π‘₯ + 2 0 π‘₯ + 1 5 = 0 2 , what is the value of 1 𝑀 + 1 𝐿 ?

  • A βˆ’ 4 3
  • B35
  • C 4 3
  • D βˆ’ 3 5
  • E βˆ’ 3 4

Q14:

Without solving the equation 3 π‘₯ βˆ’ 3 π‘₯ βˆ’ 2 = π‘₯ + 5 π‘₯ βˆ’ 7 , find the sum and the product of its roots.

  • AThe sum is 2 7 2 , and the product is 3 1 2 .
  • BThe sum is βˆ’ 2 7 , and the product is 2.
  • CThe sum is βˆ’ 2 7 2 , and the product is 3 1 2 .
  • DThe sum is 27, and the product is 31.
  • EThe sum is βˆ’ 2 7 , and the product is 31.

Q15:

Given that 𝐿 and 𝑀 are the roots of the equation π‘₯ βˆ’ 2 π‘₯ + 5 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 2 and 𝑀 2 .

  • A π‘₯ + 6 π‘₯ + 2 5 = 0 2
  • B π‘₯ + 8 π‘₯ + 2 5 = 0 2
  • C π‘₯ βˆ’ 6 π‘₯ + 2 5 = 0 2
  • D π‘₯ βˆ’ 6 π‘₯ + 1 0 = 0 2
  • E π‘₯ + 1 4 π‘₯ + 2 5 = 0 2

Q16:

Given that 𝐿 and 𝑀 are the roots of the equation 3 π‘₯ βˆ’ 6 π‘₯ + 7 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 + 𝑀 and 𝐿 𝑀 .

  • A 3 π‘₯ βˆ’ 1 3 π‘₯ + 1 4 = 0 2
  • B 3 π‘₯ + 7 π‘₯ βˆ’ 6 = 0 2
  • C 3 π‘₯ βˆ’ 6 π‘₯ + 7 = 0 2
  • D 3 π‘₯ + 1 3 π‘₯ + 1 4 = 0 2
  • E 3 π‘₯ + 6 π‘₯ + 7 = 0 2

Q17:

The roots of the equation π‘š π‘₯ + 6 𝑛 π‘₯ + 𝑙 = 0 2 , where π‘š β‰  0 , are 𝐿 and 𝑀 . Given that 𝐿 βˆ’ 𝑀 = 2 ο€Ό 1 𝐿 + 1 𝑀  , does 𝑙 ο€Ή 9 𝑛 βˆ’ π‘š 𝑙  = 3 6 𝑛 π‘š 2 2 2 2 ?

  • Ano
  • Byes

Q18:

Without solving the equation ( 7 π‘₯ + 2 ) ( 8 π‘₯ + 1 ) = 0 , find the sum of its roots.

  • A βˆ’ 2 3 5 6
  • B βˆ’ 2 3 5 6 + 9 5 6 𝑖
  • C βˆ’ 2 3 2 8
  • D βˆ’ 2 3 5 6 + 5 1 4 𝑖
  • E βˆ’ 5 6 2 3

Q19:

What is the product of the roots of the equation π‘š π‘₯ + 4 𝑛 π‘₯ + 8 𝑙 = 0 2 ?

  • A 8 𝑙 π‘š
  • B βˆ’ 4 𝑛
  • C 8 𝑙
  • D βˆ’ 4 𝑛 π‘š
  • E 4 𝑛 π‘š

Q20:

Given that √ 3 𝑖 and βˆ’ √ 3 𝑖 are the two roots of the equation π‘₯ + π‘š π‘₯ + 𝑛 = 0 2 , find the values of π‘š and 𝑛 .

  • A π‘š = 0 , 𝑛 = 3
  • B π‘š = 3 , 𝑛 = 0
  • C π‘š = 0 , 𝑛 = βˆ’ 3
  • D π‘š = 0 , 𝑛 = βˆ’ 9
  • E π‘š = 0 , 𝑛 = 9

Q21:

Find the sum and the product of the roots of the equation ( 4 π‘₯ + 1 ) ( π‘₯ + 7 ) = ( π‘₯ + 4 ) ( π‘₯ βˆ’ 8 ) without solving it.

  • AThe sum is βˆ’ 1 1 , the product is 13.
  • BThe sum is 33, the product is 3.
  • CThe sum is 11, the product is 13.
  • DThe sum is βˆ’ 3 3 , the product is 39.
  • EThe sum is 33, the product is 39.

Q22:

Given that 𝐿 and 𝑀 are the roots of the equation π‘₯ βˆ’ 1 6 π‘₯ βˆ’ 6 = 0 2 , find, in its simplest form, the quadratic equation whose roots are 𝐿 + 𝑀 and 𝐿 𝑀 .

  • A π‘₯ βˆ’ 1 0 π‘₯ βˆ’ 9 6 = 0 2
  • B π‘₯ + 1 0 π‘₯ + 9 6 = 0 2
  • C π‘₯ + 1 0 π‘₯ βˆ’ 9 6 = 0 2
  • D π‘₯ + 1 6 π‘₯ βˆ’ 6 = 0 2
  • E π‘₯ βˆ’ 1 6 π‘₯ βˆ’ 6 = 0 2

Q23:

If 𝐿 and 𝑀 are the roots of the equation π‘₯ + 2 2 π‘₯ βˆ’ 1 2 = 0 2 , what is the value of ο€Ό 𝐿 + 1 𝑀  ο€Ό 𝑀 + 1 𝐿  ?

  • A βˆ’ 1 0 1 1 2
  • B10
  • C βˆ’ 1 0
  • D 1 5 6
  • E βˆ’ 1 1 1 1 2

Q24:

Given that each root of the equation 5 π‘₯ + π‘˜ = π‘₯ + 4 2 2 is the multiplicative inverse of the other, find all possible values of π‘˜ .

  • A 3 , βˆ’ 3
  • B1
  • C9
  • D 9 , βˆ’ 9
  • E3

Q25:

The sum of the roots of the equation π‘₯ βˆ’ ( π‘˜ + 6 ) π‘₯ βˆ’ 6 π‘˜ = 0   is equal to the product of the roots of the equation 3 π‘₯ + 9 π‘˜ π‘₯ + π‘˜ = 0   . Find the possible values of π‘˜ .

  • A βˆ’ 3 or 6
  • B3 or 12
  • C3 or βˆ’ 6
  • D βˆ’ 1 or βˆ’ 1 8
  • E1 or βˆ’ 1 8
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