Unit 2 Practice Set

Simplifying radicals

Simplify. · 10 questions

a)\(\displaystyle \sqrt{12}\)

b)\(\displaystyle \sqrt{27}\)

c)\(\displaystyle \sqrt{45}\)

d)\(\displaystyle \sqrt{72}\)

e)\(\displaystyle \sqrt{98}\)

f)\(\displaystyle \sqrt{150}\)

g)\(\displaystyle \sqrt{200}\)

h)\(\displaystyle \sqrt{243}\)

i)\(\displaystyle 3\sqrt{20}\)

j)\(\displaystyle -4\sqrt{75}\)

Worked solutions for 1 →

Dividing radicals

Simplify. · 10 questions

a)\(\displaystyle \frac{\sqrt{60}}{\sqrt{5}}\)

b)\(\displaystyle \frac{\sqrt{90}}{\sqrt{10}}\)

c)\(\displaystyle \frac{\sqrt{252}}{\sqrt{7}}\)

d)\(\displaystyle \frac{4\sqrt{42}}{\sqrt{6}}\)

e)\(\displaystyle \frac{\sqrt{150}}{\sqrt{3}}\)

f)\(\displaystyle \frac{10\sqrt{66}}{\sqrt{11}}\)

g)\(\displaystyle \frac{\sqrt{96}}{\sqrt{2}}\)

h)\(\displaystyle \frac{3\sqrt{80}}{\sqrt{5}}\)

i)\(\displaystyle \frac{12\sqrt{50}}{3\sqrt{2}}\)

j)\(\displaystyle \frac{8\sqrt{105}}{\sqrt{15}}\)

Worked solutions for 2 →

Multiplying radicals

Simplify. · 10 questions

a)\(\displaystyle \sqrt{6} \times \sqrt{3}\)

b)\(\displaystyle \sqrt{10} \times \sqrt{15}\)

c)\(\displaystyle \sqrt{7} \times \sqrt{14}\)

d)\(\displaystyle 2\sqrt{3} \times \sqrt{6}\)

e)\(\displaystyle 3\sqrt{5} \times 2\sqrt{15}\)

f)\(\displaystyle 4\sqrt{2} \times 3\sqrt{10}\)

g)\(\displaystyle 2\sqrt{6} \times 5\sqrt{8}\)

h)\(\displaystyle -3\sqrt{7} \times 2\sqrt{21}\)

i)\(\displaystyle 5\sqrt{12} \times \sqrt{3}\)

j)\(\displaystyle 2\sqrt{14} \times 3\sqrt{6}\)

Worked solutions for 3 →

Radical expressions over a whole number

Simplify. · 10 questions

a)\(\displaystyle \frac{8 + 12\sqrt{3}}{4}\)

b)\(\displaystyle \frac{6 + \sqrt{72}}{3}\)

c)\(\displaystyle \frac{10 - \sqrt{50}}{5}\)

d)\(\displaystyle \frac{9 + \sqrt{27}}{3}\)

e)\(\displaystyle \frac{12 - \sqrt{48}}{4}\)

f)\(\displaystyle \frac{14 + \sqrt{98}}{7}\)

g)\(\displaystyle \frac{-6 + \sqrt{180}}{6}\)

h)\(\displaystyle \frac{20 + \sqrt{300}}{10}\)

i)\(\displaystyle \frac{15 - 6\sqrt{10}}{3}\)

j)\(\displaystyle \frac{4 + \sqrt{8}}{6}\)

Worked solutions for 4 →

Rectangle area in simplest radical form

Measurement · 5 problems

A rectangle has side lengths of \(4\sqrt{6}\) and \(2\sqrt{3}\). Express the area of the rectangle in simplest radical form.

A rectangle has side lengths of \(3\sqrt{10}\) and \(5\sqrt{2}\). Express the area of the rectangle in simplest radical form.

A rectangle has side lengths of \(\sqrt{21}\) and \(2\sqrt{7}\). Express the area of the rectangle in simplest radical form.

A rectangle has side lengths of \(5\sqrt{6}\) and \(4\sqrt{15}\). Express the area of the rectangle in simplest radical form.

A rectangle has side lengths of \(2\sqrt{14}\) and \(3\sqrt{6}\). Express the area of the rectangle in simplest radical form.

Worked solutions for 8 →

Triangle area in simplest radical form

Measurement · 5 problems

Express the area of the triangle in simplest radical form.

Worked solutions for 9 →

Completing the square

Find the value of \(c\) that will make each expression a perfect square trinomial. · 10 questions

a)\(\displaystyle x^2 + 14x + c\)

b)\(\displaystyle x^2 - 20x + c\)

c)\(\displaystyle x^2 + 9x + c\)

d)\(\displaystyle x^2 - 7x + c\)

e)\(\displaystyle x^2 + 0.8x + c\)

f)\(\displaystyle x^2 - 1.2x + c\)

g)\(\displaystyle x^2 + \frac{2}{3}x + c\)

h)\(\displaystyle x^2 - \frac{3}{5}x + c\)

i)\(\displaystyle x^2 + x + c\)

j)\(\displaystyle x^2 - 11x + c\)

Worked solutions for 10 →

Maximum and minimum values

Find the maximum or minimum value of the function and the value of \(x\) when it occurs. · 10 questions

a)\(\displaystyle y = x^2 + 8x - 5\)

b)\(\displaystyle y = x^2 - 10x + 18\)

c)\(\displaystyle y = -x^2 + 6x + 4\)

d)\(\displaystyle y = 8x - x^2 - 3\)

e)\(\displaystyle y = 2x^2 - 12x + 7\)

f)\(\displaystyle y = -3x^2 - 18x - 20\)

g)\(\displaystyle y = 5 + x^2 - 3x\)

h)\(\displaystyle y = 3x - x^2 - 1\)

i)\(\displaystyle y = 4x^2 + 6x - 2\)

j)\(\displaystyle y + 2x = -0.5x^2\)

Worked solutions for 11 →

Minimum and maximum products

Number problem · 5 problems

a) Find the minimum product of two numbers whose difference is 12.
b) What are the two numbers?

a) Find the minimum product of two numbers whose difference is 9.
b) What are the two numbers?

a) Find the minimum product of two numbers whose difference is 22.
b) What are the two numbers?

a) Find the maximum product of two numbers whose sum is 18.
b) What are the two numbers?

a) Find the maximum product of two numbers whose sum is 25.
b) What are the two numbers?

Worked solutions for 12 →

Maximum area of a triangle

Measurement · 5 problems

The sum of the base and the height of a triangle is 16 cm. Determine the maximum area of the triangle, in square centimetres.

The sum of the base and the height of a triangle is 30 cm. Determine the maximum area of the triangle, in square centimetres.

The sum of the base and the height of a triangle is 25 m. Determine the maximum area of the triangle, in square metres.

The sum of the base and the height of a triangle is 18 cm. Determine the maximum area of the triangle, in square centimetres.

The sum of the base and the height of a triangle is 13 m. Determine the maximum area of the triangle, in square metres.

Worked solutions for 13 →

Maximum enclosed area

Flower bed · 5 problems

Reena wants to fence a rectangular flower bed in her front yard. What is the maximum area she can enclose with 40 m of fencing?

Marcus has 52 m of edging to go all the way around a rectangular vegetable garden. What is the largest area the garden can have?

Priya is building a rectangular dog run against the back wall of her house, so the wall forms one side. She has 24 m of fencing for the other three sides. What is the maximum area of the run?

A farmer uses the side of a long barn as one side of a rectangular pen and has 36 m of fencing for the other three sides. What is the maximum area of the pen?

A rectangular pen is split into two equal sections by a fence parallel to two of its sides. There is 60 m of fencing in total, including the divider. What is the maximum total area?

Worked solutions for 14 →

Path of a basketball shot

Basketball · 5 problems

The path of a basketball shot can be modelled by \(h = -0.2d^2 + 1.6d + 2\), where \(h\) is the height of the ball, in metres, and \(d\) is the horizontal distance of the ball from the player, in metres.
a) Find the maximum height reached by the ball.
b) What is the horizontal distance of the ball from the player when it reaches its maximum height?
c) How far from the floor is the ball when the player releases it?

The path of a basketball shot can be modelled by \(h = -0.25d^2 + 2d + 2.2\), where \(h\) is the height of the ball, in metres, and \(d\) is the horizontal distance of the ball from the player, in metres.
a) Find the maximum height reached by the ball.
b) What is the horizontal distance of the ball from the player when it reaches its maximum height?
c) How far from the floor is the ball when the player releases it?

The path of a basketball shot can be modelled by \(h = -0.1d^2 + 1.2d + 1.9\), where \(h\) is the height of the ball, in metres, and \(d\) is the horizontal distance of the ball from the player, in metres.
a) Find the maximum height reached by the ball.
b) What is the horizontal distance of the ball from the player when it reaches its maximum height?
c) How far from the floor is the ball when the player releases it?

The path of a basketball shot can be modelled by \(h = -0.5d^2 + 3d + 2\), where \(h\) is the height of the ball, in metres, and \(d\) is the horizontal distance of the ball from the player, in metres.
a) Find the maximum height reached by the ball.
b) What is the horizontal distance of the ball from the player when it reaches its maximum height?
c) How far from the floor is the ball when the player releases it?

The path of a basketball shot can be modelled by \(h = -0.15d^2 + 1.5d + 2.1\), where \(h\) is the height of the ball, in metres, and \(d\) is the horizontal distance of the ball from the player, in metres.
a) Find the maximum height reached by the ball.
b) What is the horizontal distance of the ball from the player when it reaches its maximum height?
c) How far from the floor is the ball when the player releases it?

Worked solutions for 15 →

Solving by factoring

Solve by factoring. Check solutions. · 10 questions

a)\(\displaystyle x^2 + 11x + 28 = 0\)

b)\(\displaystyle y^2 - 2y = 35\)

c)\(\displaystyle m^2 - 49 = 0\)

d)\(\displaystyle t^2 + 12 = 7t\)

e)\(\displaystyle w^2 - 18w + 81 = 0\)

f)\(\displaystyle 9x^2 = 16\)

g)\(\displaystyle 2y^2 + 7y = -3\)

h)\(\displaystyle 3x^2 + 2 = -7x\)

i)\(\displaystyle 5t^2 = 13t + 6\)

j)\(\displaystyle 6a^2 - 11a = -3\)

Worked solutions for 19 →

The quadratic formula

Solve using the quadratic formula. Express radical answers as exact roots in simplest radical form, and as approximate roots to the nearest hundredth. · 10 questions

a)\(\displaystyle x^2 - 6x + 2 = 0\)

b)\(\displaystyle t^2 + 2t = 15\)

c)\(\displaystyle y^2 + 7y + 5 = 0\)

d)\(\displaystyle k^2 - 8k = 4\)

e)\(\displaystyle 0 = n^2 + 4n + 1\)

f)\(\displaystyle 2x^2 - 5x - 3 = 0\)

g)\(\displaystyle 3w^2 + w - 5 = 0\)

h)\(\displaystyle 2m^2 - 6m = -1\)

i)\(\displaystyle 5p^2 - 2p = 2\)

j)\(\displaystyle 4r^2 = -12r - 7\)

Worked solutions for 20 →

Adding and subtracting radicals

Simplify. · 10 questions

a)\(\displaystyle 4\sqrt{3} + 6\sqrt{3} - 2\sqrt{3}\)

b)\(\displaystyle 5\sqrt{2} - 3\sqrt{7} + 4\sqrt{7} - \sqrt{2}\)

c)\(\displaystyle \sqrt{28} + \sqrt{63}\)

d)\(\displaystyle \sqrt{50} - \sqrt{72}\)

e)\(\displaystyle \sqrt{20} - \sqrt{45} + \sqrt{80}\)

f)\(\displaystyle 3\sqrt{24} - 2\sqrt{54} + \sqrt{96}\)

g)\(\displaystyle 4\sqrt{12} - \sqrt{18} - 2\sqrt{27} + \sqrt{50}\)

h)\(\displaystyle 2\sqrt{75} + 3\sqrt{40} - \sqrt{48} - 2\sqrt{90}\)

i)\(\displaystyle \sqrt{8} + \sqrt{32} + \sqrt{98}\)

j)\(\displaystyle 3\sqrt{44} - 2\sqrt{99} + \sqrt{176}\)

Worked solutions for 26 →

Multiplying binomials with radicals

Simplify. · 10 questions

a)\(\displaystyle \sqrt{2}(\sqrt{6} + 4)\)

b)\(\displaystyle \sqrt{5}(\sqrt{15} - \sqrt{10})\)

c)\(\displaystyle 2\sqrt{3}(\sqrt{6} + 5\sqrt{2})\)

d)\(\displaystyle (\sqrt{3} + 2)(\sqrt{3} - 5)\)

e)\(\displaystyle (2\sqrt{2} + \sqrt{3})(\sqrt{2} - 4\sqrt{3})\)

f)\(\displaystyle (3\sqrt{5} + \sqrt{2})^2\)

g)\(\displaystyle (\sqrt{6} - 2)^2\)

h)\(\displaystyle (\sqrt{11} - \sqrt{5})(\sqrt{11} + \sqrt{5})\)

i)\(\displaystyle (4\sqrt{3} + 2\sqrt{2})(4\sqrt{3} - 2\sqrt{2})\)

j)\(\displaystyle (2\sqrt{5} - \sqrt{3})(3\sqrt{5} + 2\sqrt{3})\)

Worked solutions for 27 →

Rationalizing a monomial denominator

Simplify. · 10 questions

a)\(\displaystyle \frac{1}{\sqrt{7}}\)

b)\(\displaystyle \frac{3}{\sqrt{6}}\)

c)\(\displaystyle \frac{\sqrt{3}}{\sqrt{5}}\)

d)\(\displaystyle \frac{10}{\sqrt{2}}\)

e)\(\displaystyle \frac{5}{2\sqrt{5}}\)

f)\(\displaystyle \frac{6}{3\sqrt{2}}\)

g)\(\displaystyle \frac{\sqrt{2}}{5\sqrt{6}}\)

h)\(\displaystyle \frac{4\sqrt{3}}{\sqrt{8}}\)

i)\(\displaystyle \frac{7}{2\sqrt{7}}\)

j)\(\displaystyle \frac{3\sqrt{5}}{2\sqrt{15}}\)

Worked solutions for 28 →

Rationalizing with a conjugate

Simplify. · 10 questions

a)\(\displaystyle \frac{3}{\sqrt{2} + 1}\)

b)\(\displaystyle \frac{2}{\sqrt{5} - \sqrt{3}}\)

c)\(\displaystyle \frac{6}{\sqrt{7} + 2}\)

d)\(\displaystyle \frac{4}{\sqrt{6} - \sqrt{2}}\)

e)\(\displaystyle \frac{\sqrt{3}}{\sqrt{3} - 1}\)

f)\(\displaystyle \frac{2\sqrt{5}}{\sqrt{5} + 3}\)

g)\(\displaystyle \frac{\sqrt{2}}{\sqrt{2} - 4}\)

h)\(\displaystyle \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\)

i)\(\displaystyle \frac{2\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}}\)

j)\(\displaystyle \frac{3\sqrt{2} + 2\sqrt{3}}{2\sqrt{2} - \sqrt{3}}\)

Worked solutions for 29 →

Area of a square with a radical side

Measurement · 5 problems

A square has a side length of \(3 + \sqrt{2}\). Write and simplify an expression for the area of the square.

A square tile has a side length of \(5 - \sqrt{3}\) cm. Write and simplify an expression for the area of the tile.

A square patio has sides of length \(2\sqrt{5} + 1\) m. Write and simplify an expression for its area.

A square has a side length of \(\sqrt{6} - \sqrt{2}\). Write and simplify an expression for the area of the square.

A square garden plot measures \(4 + 3\sqrt{2}\) m on each side. Write and simplify an expression for its area.

Worked solutions for 30 →