Worksheet: Applications of Quadratic Equations

In this worksheet, we will practice solving word problems by forming and solving quadratic equations.

Q1:

The length of a rectangle is 2 cm more than its width. If its area is 80 cm2, what are its length and width?

  • A 40 cm, 8 cm
  • B 40 cm, 2 cm
  • C 40 cm, 10 cm
  • D 10 cm, 8 cm

Q2:

The side length of a square is π‘₯ cm, and the dimensions of a rectangle are π‘₯ cm and 2 cm. Given that the sum of their areas is 8 cm2, determine the perimeter of the square.

Q3:

A rectangular photograph measuring 6 cm by 4 cm is to be displayed in a card mount in a rectangular frame, as shown in the diagram.

Write an equation that can be used to find π‘₯ , the width of the mount, if its area is 64 cm2.

  • A ( 7 + 2 π‘₯ ) ( 5 + 2 π‘₯ ) βˆ’ 1 3 = 6 4
  • B ( 3 + 2 π‘₯ ) ( 6 βˆ’ 5 π‘₯ ) βˆ’ 2 1 = 3 2
  • C ( 9 + 5 π‘₯ ) ( 3 + 7 π‘₯ ) βˆ’ 1 5 = 1 8
  • D ( 4 + 2 π‘₯ ) ( 6 + 2 π‘₯ ) βˆ’ 2 4 = 6 4
  • E ( 5 + 2 π‘₯ ) ( 7 βˆ’ 2 π‘₯ ) βˆ’ 2 4 = 2 8

Q4:

The length of a rectangle is 26 cm more than its width. Given that its area is 120 cm2, determine its perimeter.

Q5:

The length of a rectangle is 3 cm more than double the width. The area of the rectangle is 27 cm2. Write an equation that can be used to find 𝑀 , the width of the rectangle, in centimeters.

  • A 𝑀 ( 3 𝑀 + 2 ) = 2 7
  • B 𝑀 ( 𝑀 + 3 ) = 2 7
  • C 2 𝑀 ( 2 𝑀 + 3 ) = 2 7
  • D 𝑀 ( 2 𝑀 + 3 ) = 2 7
  • E 𝑀 ( 𝑀 + 2 ) = 2 7

Q6:

The sum, 𝑆 , of the first 𝑛 consecutive integers ( 1 + 2 + 3 + 4 + β‹― + 𝑛 ) can be found using 𝑆 = 𝑛 2 ( 1 + 𝑛 ) . Starting from 1, how many consecutive integers are required to make a sum of 21?

Q7:

The height of a ball 𝑑 seconds after it was kicked from the ground is modeled by the function β„Ž , where β„Ž ( 𝑑 ) = 1 5 𝑑 βˆ’ 5 𝑑 2 .

For how long does the ball remain in the air?

For how long does the ball remain above a height of 10 m?

Q8:

Which of the following exceeds its multiplicative inverse by 1 1 3 0 ?

  • A βˆ’ 6 or 5
  • B βˆ’ 6 5 or 5 6
  • C 6 or βˆ’ 5
  • D 6 5 or βˆ’ 5 6
  • E βˆ’ 6 5 or 5 6

Q9:

A study was carried out to determine how many people in a small town were infected with the hepatitis C virus. An approximation for the number of infected people, 𝑦 , can be found using 𝑦 = βˆ’ 0 . 5 𝑛 βˆ’ 5 . 5 𝑛 + 9 3 1  , where 𝑛 is the number of years after 2006. In which year do we expect there to be no infected people?

Q10:

Find the solution set of the equation π‘₯ βˆ’ 2 6 π‘₯ 9 = βˆ’ 1 6 9  in ℝ .

  • A { 2 , 8 }
  • B  βˆ’ 2 , βˆ’ 8 9 
  • C { βˆ’ 2 , βˆ’ 8 }
  • D  2 , 8 9 
  • E  βˆ’ 2 , 8 9 

Q11:

Given that nine times the square of π‘₯ is 25, what are the possible values of π‘₯ ?

  • A 3 5
  • B 3 5 or βˆ’ 3 5
  • C5 or βˆ’ 5
  • D 5 3 or βˆ’ 5 3

Q12:

Determine the positive number whose square exceeds twice its value by 15.

Q13:

Find the positive number whose square is equal to two times the number.

Q14:

The difference between the square of Daniel’s age now and 5 times his age 2 years ago is 160. How old is Daniel now?

Q15:

At which values of π‘₯ does the graph of 𝑦 = 1 2 π‘₯ βˆ’ 8 π‘₯  cross the π‘₯ -axis?

  • A0 and 2
  • B0 and βˆ’ 2 3
  • C0 and 8 3
  • D0 and 2 3
  • E0 and βˆ’ 8 3

Q16:

When twice the square of a number is added to 1, the result is 99. What is the number?

  • A98
  • B10 or βˆ’ 1 0
  • C10
  • D7 or βˆ’ 7

Q17:

Find two numbers with a sum of 10 and a product of 9.

  • A8, 2
  • B5, 5
  • C3, 7
  • D1, 9
  • E6, 5

Q18:

A study was carried out to investigate the number of people in a town infected by norovirus. The number of people infected, 𝑦 , occurring 𝑛 years after the start of the study, can be found using the equation 𝑦 = βˆ’ 2 . 5 𝑛  βˆ’ 7 . 5 𝑛 + 9 4 2 . What was the value of 𝑛 when there were 347 people infected?

Q19:

A rocket will be launched vertically upwards with a speed of 343 m/s.

It’s height after launch can be found using 𝑠 = 3 4 3 𝑑 βˆ’ 4 . 9 𝑑 ,  where 𝑠 is the rocket’s height in meters and 𝑑 is the time after launch in seconds.

What will be the height of the rocket 6 seconds after launch?

At what times will the rocket be 2β€Žβ€‰β€Ž690.1 m above the ground?

  • AThe height after 6 seconds will be 1β€Žβ€‰β€Ž881.6 m. It will be 2β€Žβ€‰β€Ž690.1 m above the ground at 10 s and 62 s.
  • BThe height after 6 seconds will be 2β€Žβ€‰β€Ž234.4 m. It will be 2β€Žβ€‰β€Ž690.1 m above the ground at 9 s and 61 s.
  • CThe height after 6 seconds will be 2β€Žβ€‰β€Ž234.4 m. It will be 2β€Žβ€‰β€Ž690.1 m above the ground at 10 s and 62 s.
  • DThe height after 6 seconds will be 1β€Žβ€‰β€Ž881.6 m. It will be 2β€Žβ€‰β€Ž690.1 m above the ground at 9 s and 61 s.
  • EThe height after 6 seconds will be 2β€Žβ€‰β€Ž028.6 m. It will be 2β€Žβ€‰β€Ž690.1 m above the ground at 9 s and 61 s.

Q20:

Find the positive number which is 66 less than twice its square.

Q21:

Given that , , and , find .

Q22:

An interior designer bought a rectangular rug for a room. The room’s floor is a rectangle of width 8 m and length 15 m, and the rug is placed centrally, leaving a border of constant width around it. If the rug covers half of the floor’s area, how wide is the border?

Q23:

Two siblings are 3 years apart in age. Write an equation for 𝑃 , the product of their ages, in terms of π‘Ž , the age of the younger sibling.

  • A 𝑃 = π‘Ž ( π‘Ž + 2 )
  • B 𝑃 = π‘Ž ( π‘Ž βˆ’ 2 )
  • C 𝑃 = π‘Ž ( π‘Ž βˆ’ 3 )
  • D 𝑃 = π‘Ž ( π‘Ž + 3 )
  • E 𝑃 = 3 π‘Ž 

Q24:

A formula for the normal systolic blood pressure for a man age 𝐴 , measured in millimeters mercury, is given as 𝑃 = 0 . 0 0 6 𝐴 βˆ’ 0 . 0 2 𝐴 + 1 2 0 2 . Find the age, to the nearest year, of a man whose normal systolic blood pressure measures 125 mmHg.

Q25:

A falling object travels a distance given by the formula 𝑑 = 5 𝑑 + 1 6 𝑑  ft, where 𝑑 is measured in seconds. How long will it take for the object to travel 74 ft?

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