Worksheet: Applications of Quadratic Equations

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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 10 cm, 8 cm
  • C 40 cm, 10 cm
  • D 40 cm, 2 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.

Which of the following equations can be used to find π‘₯ if the area of the mount is 64 cm2?

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

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

Q6:

Which of the following exceeds its multiplicative inverse by 1130?

  • A βˆ’ 6 or 5
  • B 6 5 or βˆ’56
  • C βˆ’ 6 5 or 56
  • D6 or βˆ’5
  • E βˆ’ 6 5 or 56

Q7:

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

  • A 5 3 or βˆ’53
  • B5 or βˆ’5
  • C 3 5
  • D 3 5 or βˆ’35

Q8:

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

Q9:

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

Q10:

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?

Q11:

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

  • A7 or βˆ’7
  • B10 or βˆ’10
  • C10
  • D98

Q12:

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

  • A8, 2
  • B6, 5
  • C1, 9
  • D5, 5
  • E3, 7

Q13:

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

Q14:

Given that οƒ«πΆπ·βˆ©βƒ–οƒ©οƒ©οƒ©οƒ©βƒ—π΄π΅={𝐢}, π‘šβˆ π΅πΆπ·=ο€Ήπ‘§ο…οŠ¨βˆ˜, and π‘šβˆ π΄πΆπ·=(41𝑧)∘, find 𝑧.

Q15:

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?

Q16:

Point 𝐾 lies between points 𝐽 and 𝐿. If 𝐽𝐾=π‘₯+8π‘₯, 𝐾𝐿=3π‘₯βˆ’2, and 𝐽𝐿=40, find the length of 𝐽𝐾.

Q17:

The sum of the squares of two positive real numbers is 542. Given that one of them is 18, find the value of the other one.

  • A √ 8 6 6
  • B βˆ’ √ 8 6 6 or √866
  • C 2 √ 1 3 1
  • D √ 2 1 8
  • E βˆ’ √ 2 1 8 or √218

Q18:

The height of a right triangle is 2 m less than three times its base length. Its area is 4 m2. Write an equation that can be used to find the base length of the triangle, 𝑏, in meters.

  • A 𝑏 2 ( 3 𝑏 βˆ’ 2 ) = 4
  • B 𝑏 2 ( 2 𝑏 βˆ’ 3 ) = 4
  • C 2 𝑏 ( 2 𝑏 βˆ’ 3 ) = 4
  • D 2 𝑏 ( 3 𝑏 βˆ’ 2 ) = 4
  • E 𝑏 2 ( 3 𝑏 + 2 ) = 4

Q19:

Mr. William tells his math class, β€œWhen I subtract a positive number from the square of the number, the result is the same as when I triple the sum of the number and four.”

What was Mr. William’s number?

Q20:

Consider the following figure.

Write an equation for 𝐴, the area of the rectangle, in terms of 𝑏. Simplify your equation so that there are no brackets.

  • A 𝐴 = 𝑏 βˆ’ 2 𝑏 + 3 
  • B 𝐴 = 𝑏 βˆ’ 3 𝑏 + 2 
  • C 𝐴 = 𝑏 βˆ’ 2 𝑏 βˆ’ 3 
  • D 𝐴 = 𝑏 + 2 𝑏 + 3 
  • E 𝐴 = 𝑏 + 2 𝑏 βˆ’ 3 

If the value of 𝑏 is 9, what is the area of the rectangle?

If the area of the rectangle is 32, what is the value of 𝑏?

Q21:

What is the perimeter of a rectangle whose length is 7 cm more than its width and whose area is 78 cm2?

  • A18 cm
  • B19 cm
  • C24 cm
  • D17 cm
  • E38 cm

Q22:

The diagram shows a rectangular prism, where the area of its net is 580. Find the value of π‘₯.

Q23:

A rectangle has a perimeter of 18, where its sides have dimensions 7𝑦 and 𝑦+1. Find the ratio of its longer side to its shorter one.

  • A 2 ∢ 7
  • B 6 5 ∢ 5 6
  • C 8 ∢ 7
  • D 7 ∢ 2
  • E 5 6 ∢ 6 5

Q24:

A right-angled triangle has area 150 cm2. If the lengths of its perpendicular sides are (π‘₯+13) cm and (3π‘₯+14) cm, what is its perimeter?

Q25:

The diagram shows a trapezoid and a rectangle.

Write an expression for the area of the rectangle.

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

Write an expression for the area of the trapezoid.

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

If the trapezoid and the rectangle have the same area, find the value of π‘₯ using a suitable equation

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