Worksheet: Quadratic Formula

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

Q1:

Solve the equation − 𝑥 + 7 𝑥 + 1 = 0 2 .

  • A 4 9 − √ 1 1 2 , 4 9 + √ 1 1 2
  • B 7 − √ 1 1 2 , 7 + √ 1 1 2
  • C 7 − √ 5 1 2 , 7 + √ 5 1 2
  • D 7 − √ 5 3 2 , 7 + √ 5 3 2
  • E − 7 + √ 5 3 2 , − 7 − √ 5 3 2

Q2:

Find the solution set of the equation 3 𝑥 − 2 ( 7 − 𝑥 ) = 0 , giving values to one decimal place.

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

Q3:

Find the solution set of the equation 𝑥 − 8 𝑥 − 2 = 9 𝑥 + 8 , giving values correct to three decimal places.

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

Q4:

Find the solution set of the equation 5 𝑥 − 7 𝑥 − 3 2 = 0 , giving values to three decimal places.

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

Q5:

Find the solution set of the equation 1 8 𝑥 + 5 𝑥 = 1 , giving values to three decimal places.

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

Q6:

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

  • A { 3 , 5 }
  • B − 3 4 , − 5 4
  • C { − 3 , − 5 }
  • D 3 4 , 5 4
  • E 3 4 , − 5 4

Q7:

Find the solution set of the equation 5 𝑥 ( 𝑥 − 6 ) − 3 ( 𝑥 + 4 ) + 3 = 0 , giving values to one decimal place.

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

Q8:

Find the solution set of the equation 2 𝑥 − 5 = 6 𝑥 , giving values to three decimal places.

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

Q9:

Given that 𝑥 = − 1 is one of the roots of the equation − 3 𝑥 − 9 𝑥 + 𝑘 = 0 , find the other root and the value of 𝑘 .

  • A 𝑥 = 4 , 𝑘 = − 4
  • B 𝑥 = − 2 , 𝑘 = 2
  • C 𝑥 = 4 , 𝑘 = 1 2
  • D 𝑥 = − 2 , 𝑘 = − 6
  • E 𝑥 = − 2 , 𝑘 = 6

Q10:

Find the solution set of the equation ( 𝑥 − 5 ) − ( 𝑥 − 5 ) − 3 = 0 , giving values to one decimal place.

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

Q11:

The sum of the roots of the equation 4 𝑥 + 𝑘 𝑥 − 4 = 0 is − 1 . Find the value of 𝑘 and the solution set of the equation.

  • A4, − 1 − √ 2 2 , − 1 + √ 2 2
  • B − 4 , 1 − √ 5 2 , 1 + √ 5 2
  • C − 4 , 1 + √ 2 2 , 1 − √ 2 2
  • D4, − 1 − √ 5 2 , − 1 + √ 5 2

Q12:

Given that 𝑥 = − 2 is a root of the equation 𝑥 − 4 𝑚 𝑥 − 𝑚 − 6 = 0 , find the set of possible values of 𝑚 ?

  • A 3 2 − 2 √ 3 , 3 2 + 2 √ 3
  • B 4 − 2 √ 3 , 4 + 2 √ 3
  • C 8 − √ 6 6 4 0 , 8 + √ 6 6 4 0
  • D 4 − √ 2 6 , 4 + √ 2 6
  • E − 8 + 2 √ 2 6 2 0 , − 8 − 2 √ 2 6 2 0

Q13:

Find the solution set of the equation 3 𝑥 + 3 − 4 𝑥 − 3 = 3 in ℝ , giving values to one decimal place.

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

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 − 3 𝑥 − 𝑥 + 1 2 = 0 in ℝ , giving values to one decimal place.

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

Q16:

If 𝐿 and 𝑀 are the roots of the equation 𝑥 + 2 4 𝑥 − 6 = 0 , what is the value of 𝐿 + 2 4 𝐿 − 1 7 ?

Q17:

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

  • A { 5 . 8 , − 5 . 8 }
  • B { − 0 . 2 , 0 . 2 }
  • C { − 0 . 3 , 0 . 3 }
  • D ∅

Q18:

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

  • A 𝑥 = 1 . 5 4 , 𝑥 = − 4 . 5 4 4
  • B 𝑥 = − 0 . 7 5 + 1 . 7 1 𝑖 , 𝑥 = − 0 . 7 5 − 1 . 7 1 𝑖
  • C 𝑥 = 7 6 , 𝑥 = − 7 6
  • D 𝑥 = − 2 . 7 7 , 𝑥 = 1 . 2 7
  • E 𝑥 = 1 . 2 7

Q19:

Use the quadratic formula to find all the values of 𝑎 for which the roots of the equation 2 𝑥 + 𝑎 𝑥 − 7 = 0 2 differ by exactly 10. Give your answers correct to two decimal places if necessary.

  • A4, − 4
  • B9.27, − 9 . 2 7
  • C9.79, − 9 . 7 9
  • D18.55, − 1 8 . 5 5
  • E20, − 2 0

Q20:

Find the solution set of the equation ( 𝑥 − 2 3 ) − 6 𝑥 = 0 , giving values to three decimal places.

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

Q21:

Find the solution set in ℝ of the equation 4 𝑥 − 2 𝑥 = 1 , giving values to one decimal place.

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

Q22:

Using the quadratic formula, find all the solutions to 𝑥 − 1 0 𝑥 + 1 = 0 4 2 .

  • A 𝑥 = 5 + √ 3 4 , 𝑥 = − 5 + √ 3 4 , 𝑥 = 5 − 2 √ 3 4 , 𝑥 = − 5 − √ 3 4
  • B 𝑥 = 5 + √ 1 5 , 𝑥 = − 5 + √ 1 5 , 𝑥 = 5 − 2 √ 1 5 , 𝑥 = − 5 − √ 1 5
  • C 𝑥 = 5 + 2 √ 6 , 𝑥 = 5 − 2 √ 6
  • D 𝑥 = 5 + 2 √ 6 , 𝑥 = − 5 + 2 √ 6 , 𝑥 = 5 − 2 √ 6 , 𝑥 = − 5 − 2 √ 6
  • E 𝑥 = 5 + √ 1 5 , 𝑥 = 5 − √ 1 5

Q23:

Find the solution set of the equation 𝑥 + 4 6 𝑥 = 2 2 , giving values to three decimal places.

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

Q24:

Find the solution set of 𝑥 − 6 ( 𝑥 − 1 ) = 2 in ℝ , giving values to two decimal places.

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

Q25:

Find the solution set of the equation 𝑥 ( 𝑥 − 3 6 ) = 2 1 , giving values to three decimal places.

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

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