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?

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

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.

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

Q6:

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

Q7:

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

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

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

  • 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?

Q21:

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

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.

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

Q24:

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?

Q25:

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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