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Lesson: Quadratic Formula

In this lesson, we will learn how to solve quadratic equations using the quadratic formula.

Video

17:06

Sample Question Videos

01:37
05:36
04:46

Worksheet • 25 Questions • 3 Videos

Q1:

Solve the equation βˆ’ π‘₯ + 7 π‘₯ + 1 = 0 2 .

  • A  7 βˆ’ √ 5 3 2 , 7 + √ 5 3 2 
  • B  βˆ’ 7 + √ 5 3 2 , βˆ’ 7 βˆ’ √ 5 3 2 
  • C  7 βˆ’ √ 1 1 2 , 7 + √ 1 1 2 
  • D  7 βˆ’ √ 5 1 2 , 7 + √ 5 1 2 
  • E  4 9 βˆ’ √ 1 1 2 , 4 9 + √ 1 1 2 

Q2:

Find the solution set of the equation 3 π‘₯ βˆ’ 2 ( 7 βˆ’ π‘₯ ) = 0 2 , giving values to one decimal place.

  • A { 1 . 9 , βˆ’ 2 . 5 }
  • B { βˆ’ 1 . 9 , βˆ’ 2 . 5 }
  • C { 3 . 7 , βˆ’ 5 . 0 }
  • D { βˆ’ 3 . 7 , βˆ’ 5 . 0 }

Q3:

Find the solution set of the equation π‘₯ βˆ’ 8 π‘₯ βˆ’ 2 = 9 π‘₯ + 8 2 , giving values correct to three decimal places.

  • A { 1 7 . 5 6 9 , βˆ’ 0 . 5 6 9 }
  • B { 0 . 5 6 9 , βˆ’ 1 7 . 5 6 9 }
  • C { 0 . 2 8 5 , βˆ’ 8 . 7 8 5 }
  • D { 8 . 7 8 5 , βˆ’ 0 . 2 8 5 }

Q4:

Find the solution set of the equation 5 π‘₯ βˆ’ 7 π‘₯ βˆ’ 3 2 = 0 2 , giving values to three decimal places.

  • A { 3 . 3 2 5 , βˆ’ 1 . 9 2 5 }
  • B { 3 . 3 2 5 , βˆ’ 3 . 8 5 0 }
  • C { βˆ’ 3 . 3 2 5 , 1 . 9 2 5 }
  • D { 6 . 6 5 0 , βˆ’ 3 . 8 5 0 }
  • E { βˆ’ 6 . 6 5 0 , βˆ’ 3 . 8 5 0 }

Q5:

Find the solution set of the equation 1 8 π‘₯ + 5 π‘₯ = 1 2 , giving values to three decimal places.

  • A { 7 . 4 2 4 , βˆ’ 2 . 4 2 4 }
  • B { βˆ’ 7 . 4 2 4 , 2 . 4 2 4 }
  • C { βˆ’ 7 . 4 2 4 , 4 . 8 4 9 }
  • D { βˆ’ 1 4 . 8 4 9 , 4 . 8 4 9 }

Q6:

Find the solution set of 1 6 𝑧 = 3 2 𝑧 βˆ’ 1 5 2 in ℝ .

  • A  3 4 , 5 4 
  • B  3 4 , βˆ’ 5 4 
  • C  βˆ’ 3 4 , βˆ’ 5 4 
  • D { βˆ’ 3 , βˆ’ 5 }
  • E { 3 , 5 }

Q7:

Find the solution set of the equation 5 π‘₯ ( π‘₯ βˆ’ 6 ) βˆ’ 3 ( π‘₯ + 4 ) + 3 = 0 , giving values to one decimal place.

  • A { 6 . 9 , βˆ’ 0 . 3 }
  • B { βˆ’ 6 . 9 , βˆ’ 0 . 3 }
  • C { 1 3 . 7 , βˆ’ 0 . 5 }
  • D { βˆ’ 1 3 . 7 , βˆ’ 0 . 5 }

Q8:

Find the solution set of the equation 2 π‘₯ βˆ’ 5 = 6 π‘₯ , giving values to three decimal places.

  • A { 3 . 3 8 6 , βˆ’ 0 . 8 8 6 }
  • B { 0 . 8 8 6 , βˆ’ 3 . 3 8 6 }
  • C { 1 . 7 7 2 , βˆ’ 6 . 7 7 2 }
  • D { 6 . 7 7 2 , βˆ’ 1 . 7 7 2 }

Q9:

Given that π‘₯ = βˆ’ 1 is one of the roots of the equation βˆ’ 3 π‘₯ βˆ’ 9 π‘₯ + π‘˜ = 0 2 , find the other root and the value of π‘˜ .

  • A π‘₯ = βˆ’ 2 , π‘˜ = βˆ’ 6
  • B π‘₯ = βˆ’ 2 , π‘˜ = 6
  • C π‘₯ = βˆ’ 2 , π‘˜ = 2
  • D π‘₯ = 4 , π‘˜ = 1 2
  • E π‘₯ = 4 , π‘˜ = βˆ’ 4

Q10:

Find the solution set of the equation ( π‘₯ βˆ’ 5 ) βˆ’ ( π‘₯ βˆ’ 5 ) βˆ’ 3 = 0 2 , giving values to one decimal place.

  • A { 7 . 3 , 3 . 7 }
  • B { βˆ’ 7 . 3 , 3 . 7 }
  • C { 1 4 . 6 , 7 . 4 }
  • D { βˆ’ 1 4 . 6 , 7 . 4 }

Q11:

The sum of the roots of the equation 4 π‘₯ + π‘˜ π‘₯ βˆ’ 4 = 0 2 is βˆ’ 1 . Find the value of π‘˜ and the solution set of the equation.

  • A4,  βˆ’ 1 βˆ’ √ 5 2 , βˆ’ 1 + √ 5 2 
  • B βˆ’ 4 ,  1 βˆ’ √ 5 2 , 1 + √ 5 2 
  • C βˆ’ 4 ,  1 + √ 2 2 , 1 βˆ’ √ 2 2 
  • D4,  βˆ’ 1 βˆ’ √ 2 2 , βˆ’ 1 + √ 2 2 

Q12:

Given that is a root of the equation , find the set of possible values of ?

  • A
  • B
  • C
  • D
  • E

Q13:

Find the solution set of the equation 3 π‘₯ + 3 βˆ’ 4 π‘₯ βˆ’ 3 = 3 in ℝ , giving values to one decimal place.

  • A { 1 . 3 , βˆ’ 1 . 6 }
  • B { 1 . 6 , βˆ’ 1 . 3 }
  • C { 1 . 4 , βˆ’ 1 . 4 }
  • D { 7 . 5 , βˆ’ 9 . 5 }

Q14:

The dimensions of a rectangle are 5 m and 12 m. When both dimensions are increased by a given amount, the area of the rectangle will double. What is the amount?

Q15:

Find the solution set of the equation in , giving values to one decimal place if necessary.

  • A
  • B
  • C
  • D

Q16:

If 𝐿 and 𝑀 are the roots of the equation π‘₯ + 2 4 π‘₯ βˆ’ 6 = 0 2 , what is the value of 𝐿 + 2 4 𝐿 βˆ’ 1 7 2 ?

Q17:

Find the solution set of π‘₯ + 1 7 π‘₯ = 6 in ℝ , giving values to one decimal place.

  • A βˆ…
  • B { βˆ’ 0 . 2 , 0 . 2 }
  • C { βˆ’ 0 . 3 , 0 . 3 }
  • D { 5 . 8 , βˆ’ 5 . 8 }

Q18:

By using the quadratic formula, solve the equation 2 π‘₯ + 3 π‘₯ = 7 2 . Give your answers correct to two decimal places.

  • A π‘₯ = βˆ’ 2 . 7 7 , π‘₯ = 1 . 2 7
  • B π‘₯ = 1 . 2 7
  • C π‘₯ = βˆ’ 0 . 7 5 + 1 . 7 1 𝑖 , π‘₯ = βˆ’ 0 . 7 5 βˆ’ 1 . 7 1 𝑖
  • D π‘₯ = 7 6 , π‘₯ = βˆ’ 7 6
  • E π‘₯ = 1 . 5 4 , π‘₯ = βˆ’ 4 . 5 4 4

Q19:

Use the quadratic formula to find all the values of π‘Ž for which the roots of the equation 2 π‘₯ + π‘Ž π‘₯ βˆ’ 7 = 0 2 differ by exactly 10. Give your answers correct to two decimal places if necessary.

  • A18.55, βˆ’ 1 8 . 5 5
  • B20, βˆ’ 2 0
  • C9.27, βˆ’ 9 . 2 7
  • D9.79, βˆ’ 9 . 7 9
  • E4, βˆ’ 4

Q20:

Find the solution set of the equation ( π‘₯ βˆ’ 2 3 ) βˆ’ 6 π‘₯ = 0 2 , giving values to three decimal places.

  • A { 3 8 . 1 2 4 , 1 3 . 8 7 6 }
  • B { 3 8 . 1 2 4 , 2 7 . 7 5 1 }
  • C { βˆ’ 3 8 . 1 2 4 , 1 3 . 8 7 6 }
  • D { 7 6 . 2 4 9 , 2 7 . 7 5 1 }
  • E { βˆ’ 7 6 . 2 4 9 , 2 7 . 7 5 1 }

Q21:

Find the solution set in ℝ of the equation 4 π‘₯ βˆ’ 2 π‘₯ = 1 , giving values to one decimal place.

  • A { 0 . 8 , βˆ’ 0 . 6 }
  • B { 0 . 6 , βˆ’ 0 . 8 }
  • C { 6 . 7 , βˆ’ 4 . 7 }
  • D { 4 . 7 , βˆ’ 6 . 7 }

Q22:

Using the quadratic formula, find all the solutions to π‘₯ βˆ’ 1 0 π‘₯ + 1 = 0 4 2 .

  • A π‘₯ =  5 + 2 √ 6 , π‘₯ = βˆ’  5 + 2 √ 6 , π‘₯ =  5 βˆ’ 2 √ 6 , π‘₯ = βˆ’  5 βˆ’ 2 √ 6
  • B π‘₯ = 5 + √ 1 5 , π‘₯ = 5 βˆ’ √ 1 5
  • C π‘₯ =  5 + √ 1 5 , π‘₯ = βˆ’  5 + √ 1 5 , π‘₯ =  5 βˆ’ 2 √ 1 5 , π‘₯ = βˆ’  5 βˆ’ √ 1 5
  • D π‘₯ = 5 + 2 √ 6 , π‘₯ = 5 βˆ’ 2 √ 6
  • E π‘₯ =  5 + √ 3 4 , π‘₯ = βˆ’  5 + √ 3 4 , π‘₯ =  5 βˆ’ 2 √ 3 4 , π‘₯ = βˆ’  5 βˆ’ √ 3 4

Q23:

Find the solution set of the equation π‘₯ + 4 6 π‘₯ = 2 2 , giving values to three decimal places.

  • A { 1 9 . 6 6 0 , 2 . 3 4 0 }
  • B { 1 9 . 6 6 0 , 4 . 6 7 9 }
  • C { βˆ’ 1 9 . 6 6 0 , 2 . 3 4 0 }
  • D { 3 9 . 3 2 1 , 4 . 6 7 9 }
  • E { βˆ’ 3 9 . 3 2 1 , 4 . 6 7 9 }

Q24:

Find the solution set of π‘₯ βˆ’ 6 ( π‘₯ βˆ’ 1 ) = 2 2 in ℝ , giving values to two decimal places.

  • A { 5 . 2 4 , 0 . 7 6 }
  • B { 6 . 6 1 , βˆ’ 0 . 6 1 }
  • C { βˆ’ 5 . 2 4 , βˆ’ 0 . 7 6 }
  • D { βˆ’ 1 0 . 4 7 , βˆ’ 1 . 5 3 }
  • E { 1 0 . 4 7 , 1 . 5 3 }

Q25:

Find the solution set of the equation , giving values to three decimal places if necessary.

  • A
  • B
  • C
  • D
  • E
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