Worksheet: Applications of Quadratic Equations

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


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

  • A40 cm, 8 cm
  • B10 cm, 8 cm
  • C40 cm, 10 cm
  • D40 cm, 2 cm


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.


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+2π‘₯)(6βˆ’5π‘₯)βˆ’21=32
  • B(9+5π‘₯)(3+7π‘₯)βˆ’15=18
  • C(7+2π‘₯)(5+2π‘₯)βˆ’13=64
  • D(5+2π‘₯)(7βˆ’2π‘₯)βˆ’24=28
  • E(4+2π‘₯)(6+2π‘₯)βˆ’24=64


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


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.

  • A𝑀(3𝑀+2)=27
  • B2𝑀(2𝑀+3)=27
  • C𝑀(𝑀+3)=27
  • D𝑀(2𝑀+3)=27
  • E𝑀(𝑀+2)=27


Which of the following exceeds its multiplicative inverse by 1130?

  • Aβˆ’6 or 5
  • B65 or βˆ’56
  • Cβˆ’65 or 56
  • D6 or βˆ’5
  • Eβˆ’65 or 56


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

  • A53 or βˆ’53
  • B5 or βˆ’5
  • C35
  • D35 or βˆ’35


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


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


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?


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


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

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


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


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.


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?


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


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√866
  • Bβˆ’βˆš866 or √866
  • C2√131
  • D√218
  • Eβˆ’βˆš218 or √218


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.

  • A𝑏2(3π‘βˆ’2)=4
  • B𝑏2(2π‘βˆ’3)=4
  • C2𝑏(2π‘βˆ’3)=4
  • D2𝑏(3π‘βˆ’2)=4
  • E𝑏2(3𝑏+2)=4


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?


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.

  • A𝐴=π‘βˆ’2𝑏+3
  • B𝐴=π‘βˆ’3𝑏+2
  • C𝐴=π‘βˆ’2π‘βˆ’3
  • D𝐴=𝑏+2𝑏+3
  • E𝐴=𝑏+2π‘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?


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


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


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.

  • A2∢7
  • B65∢56
  • C8∢7
  • D7∢2
  • E56∢65


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


The diagram shows a trapezoid and a rectangle.

Write an expression for the area of the rectangle.

  • A(2π‘₯+1)(π‘₯βˆ’9)
  • B2(2π‘₯+1)(π‘₯βˆ’9)
  • C(2π‘₯βˆ’1)(π‘₯+8)
  • D(2π‘₯+3)(π‘₯βˆ’7)
  • E2((2π‘₯+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)(2π‘₯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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