Worksheet: Applications of 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 cm, and the dimensions of a rectangle are 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 if the area of the mount is 64 cm²?

A
B
C
D
E

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 , the width of the rectangle, in centimeters.

A
B
C
D
E

Q6:

Which of the following exceeds its multiplicative inverse by ?

A or 5
B or
C or
D6 or
E or

Q7:

Given that nine times the square of is 25, what are the possible values of ?

A or
B5 or
C
D or

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
B10 or
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 , , and , find .

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 lies between points and . If , , and , find the length of .

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
B or
C
D
E or

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, , in meters.

A
B
C
D
E

Q19:

Mr. William tells his math class, “When 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 , the area of the rectangle, in terms of . Simplify your equation so that there are no brackets.

A
B
C
D
E

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

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

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 .

Q23:

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

A
B
C
D
E

Q24:

A right triangle has area 150 cm². If the lengths of its perpendicular sides are cm and cm, what is its perimeter?

Q25:

The diagram shows a trapezoid and a rectangle.

Write an expression for the area of the rectangle.

A
B
C
D
E

Write an expression for the area of the trapezoid.

A
B
C
D
E

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