Worksheet: Solving Quadratic Equations: Quadratic Formula

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In this worksheet, we will practice solving quadratic equations using the quadratic formula.

Q1:

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

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

Q2:

Find the solution set of the equation 3π‘₯βˆ’2(7βˆ’π‘₯)=0, 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’8π‘₯βˆ’2=9π‘₯+8, 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 { 8 . 7 8 5 , βˆ’ 0 . 2 8 5 }
  • D { 0 . 2 8 5 , βˆ’ 8 . 7 8 5 }

Q4:

Find the solution set of the equation 5π‘₯βˆ’7π‘₯βˆ’32=0, giving values to three decimal places.

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

Q5:

Find the solution set of the equation 18π‘₯+5π‘₯=1, giving values to three decimal places.

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

Q6:

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

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

Q7:

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

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

Q8:

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

  • A4, ο―βˆ’1βˆ’βˆš22,βˆ’1+√22
  • B βˆ’ 4 ,  1 βˆ’ √ 5 2 , 1 + √ 5 2 
  • C βˆ’ 4 ,  1 + √ 2 2 , 1 βˆ’ √ 2 2 
  • D4, ο―βˆ’1βˆ’βˆš52,βˆ’1+√52

Q9:

Given that π‘₯=βˆ’2 is a root of the equation π‘₯βˆ’4π‘šπ‘₯βˆ’ο€Ήπ‘šβˆ’6=0, find the set of possible values of π‘š?

  • A  4 βˆ’ 2 √ 3 , 4 + 2 √ 3 
  • B  3 2 βˆ’ 2 √ 3 , 3 2 + 2 √ 3 
  • C  βˆ’ 8 + 2 √ 2 6 2 0 , βˆ’ 8 βˆ’ 2 √ 2 6 2 0 
  • D  4 βˆ’ √ 2 6 , 4 + √ 2 6 
  • E  8 βˆ’ √ 6 6 4 0 , 8 + √ 6 6 4 0 

Q10:

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

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

Q11:

Find the solution set of the equation βˆ’3π‘₯βˆ’π‘₯+12=0 in ℝ, giving values to one decimal place.

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

Q12:

Find the solution set of π‘₯+17π‘₯=6 in ℝ, giving values to one decimal place.

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

Q13:

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

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

Q14:

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

  • A20, βˆ’20
  • B9.27, βˆ’9.27
  • C4, βˆ’4
  • D9.79, βˆ’9.79
  • E18.55, βˆ’18.55

Q15:

Find the solution set of the equation (π‘₯βˆ’23)βˆ’6π‘₯=0, giving values to three decimal places.

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

Q16:

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

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

Q17:

Using the quadratic formula, find all the solutions to π‘₯βˆ’10π‘₯+1=0οŠͺ.

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

Q18:

Find the solution set of the equation π‘₯+46π‘₯=22, giving values to three decimal places.

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

Q19:

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

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

Q20:

Find the solution set of the equation π‘₯(π‘₯βˆ’36)=21, giving values to three decimal places.

  • A { 3 6 . 5 7 4 , βˆ’ 0 . 5 7 4 }
  • B { 3 6 . 5 7 4 , βˆ’ 1 . 1 4 8 }
  • C { 7 3 . 1 4 8 , βˆ’ 1 . 1 4 8 }
  • D { βˆ’ 3 6 . 5 7 4 , 0 . 5 7 4 }
  • E { βˆ’ 7 3 . 1 4 8 , βˆ’ 1 . 1 4 8 }

Q21:

Find the solution set of the equation 6π‘₯βˆ’8π‘₯+1=0, giving values to two decimal places.

  • A { βˆ’ 2 . 3 9 , βˆ’ 0 . 2 8 }
  • B { 2 . 3 9 , 0 . 2 8 }
  • C { βˆ’ 1 . 1 9 , βˆ’ 0 . 1 4 }
  • D { 1 . 1 9 , 0 . 1 4 }

Q22:

Find the solution set of the equation π‘₯=βˆ’18π‘₯+6, giving values to three decimal places.

  • A { 1 8 . 3 2 7 , βˆ’ 0 . 3 2 7 }
  • B { 0 . 3 2 7 , βˆ’ 1 8 . 3 2 7 }
  • C { 0 . 1 6 4 , βˆ’ 9 . 1 6 4 }
  • D { 9 . 1 6 4 , βˆ’ 0 . 1 6 4 }

Q23:

Find the solution set of the equation 9π‘₯βˆ’4π‘₯=20, giving values to three decimal places.

  • A { 3 . 4 5 9 , βˆ’ 2 . 5 7 0 }
  • B { 1 . 7 2 9 , βˆ’ 1 . 2 8 5 }
  • C { 2 . 5 7 0 , βˆ’ 3 . 4 5 9 }
  • D { 1 . 2 8 5 , βˆ’ 1 . 7 2 9 }

Q24:

Find the solution set of the equationβˆ’5βˆ’5π‘₯=1π‘₯ in ℝ, giving values to one decimal place.

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

Q25:

Find the solution set of the equation βˆ’π‘₯βˆ’1βˆ’2π‘₯βˆ’6=15 in ℝ, giving values to one decimal place.

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

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