Chapter
Chapter 6
Section
Solutions 42 Videos

Graph the parabola by completing the square. Label the vertex, the axis of symmetry, and two other points.

\displaystyle y = x^2 + 6x + 4 

Q1a

Graph the parabola by completing the square. Label the vertex, the axis of symmetry, and two other points.

\displaystyle y = -x^2 + 8x -3 

Q1b

Graph the parabola by completing the square. Label the vertex, the axis of symmetry, and two other points.

\displaystyle y = 3x^2 + 24x + 10 

Q1c

Solve each quadratic equation by factoring.

\displaystyle 9y^2 - 1= 0 

Q2b

Solve each quadratic equation by factoring.

\displaystyle x^2 = 3x + 10 

Q2c

Solve each quadratic equation by factoring.

\displaystyle 9b^2 -12b + 4 = 0 

Q2d

Solve each quadratic equation by factoring.

\displaystyle 3x^2 + 13x = 10 

Q2e

Solve each quadratic equation by factoring.

\displaystyle 6m^2 + 30m = 0 

Q2f

Solve each quadratic equation by factoring.

\displaystyle 3x^2 +13x = 10 

Q2g

Solve each quadratic equation by factoring.

\displaystyle 3d + 1 = -4d^2 

Q2h

Find the x-intercepts, axis of symmetry, and vertex of each parabola. Then, graph the relation, labelling it fully.

\displaystyle y = x^2 -10x + 25 

Q4c

\displaystyle 4x^2 -11x - 3= 0 

Q5a

\displaystyle x^2 +5x = 7 

Q5b

\displaystyle 9x^2=30x -25 

Q5c

\displaystyle 7k^2-9k+3= 0 

Q5d

\displaystyle 4s^2-9s=-3 

Q5e

\displaystyle 3t^2-7=t 

Q5f

Use an appropriate method to find the roots of each equation.

\displaystyle 3x^2+12x+ 6 = 0 

Q6a

Use an appropriate method to find the roots of each equation.

\displaystyle x^2 -8x+3 = 0 

Q6b

Use an appropriate method to find the roots of each equation.

\displaystyle 4m^2 -10=0 

Q6c

Use an appropriate method to find the roots of each equation.

\displaystyle -5x^2 +10x=5 

Q6d

Use an appropriate method to find the roots of each equation.

\displaystyle (k - 5)^2=16 

Q6e

Use an appropriate method to find the roots of each equation.

\displaystyle \frac{x^2}{2} + x + \frac{1}{2} = 0 

Q6f

Use an appropriate method to find the roots of each equation.

\displaystyle 2(m-1)^2 =(m+2)(m+1) 

Q6g

Use an appropriate method to find the roots of each equation.

\displaystyle (5x +2)(3x - 1)=4x^2+5 

Q6h

Compare the axis of symmetry of

\displaystyle y = 3x^2+18x -17  to that of \displaystyle y = 3x^2 + 18x +1 . Explain your findings.

Q7

The path of a firework is modelled using the equation h = -5d^2 + 20d+ 1, where h is the height, in metres, above the ground and d is the horizontal distance, in metres. What is the maximum height of the firework?

Q8

A parabola is defined by the equation \displaystyle y = -2(x + 1)^2 + 18 .

a) Without solving, explain how you can tell how many x-intercepts there are.

b) Given the current form of the equation, what is the easiest way to find the x-intercepts?

c) How far apart are the x-intercepts?

Q9

Write a quadratic equation in the form ax2 + bx + c = 0 for each situation, where a, b, and c are integers.

• The roots of the equation are 5 and -3.
Q10a

Write a quadratic equation in the form ax2 + bx + c = 0 for each situation, where a, b, and c are integers.

• The roots of the equation are \frac{1}{2} and \frac{3}{5}.
Q10b

An equipment storage shed has a parabolic cross section modelled by the relation \displaystyle h = - d^2 + 4d , where h is the height, in metres, and d is the horizontal distance, in metres, from one edge of the shed.

a) How wide and how tall is the shed?

b) Sketch the graph.

c) For what values of d is the relation valid? Explain.

Q11

The cost, in dollars, of operating a machine per day is given by the formula \displaystyle C . = 3t^2-96t+1014 , where t is the time the machine operates, in hours. What is the minimum cost of running the machine? For how many hours must the machine run to reach this minimum cost?

Q12

A triangle has base 2x + 1 and height 6x - 3. What value of x would give an area of 240 m^2? Round to the nearest hundredth.

Q13

The relation \displaystyle d = 0.0052s^2 + 0.13s  models the stopping distance, d, in metres, of a car travelling at a speed of s, in kilometres per hour, when the driver brakes hard. At what speed was a car travelling if its stopping distance is 20m? Round to the nearest tenth?

Q14

Van dives off a 4-m springboard. His height, h, in metres, above the surface of the water is defined by the relation h = -d^2 + 3d + 4, where d is his horizontal distance, in metres, from the end of the board.

a) Determine the zeros of the relation.

b) Sketch a graph of the relation.

c) For what values of d is the relation valid?

d) What is Van’s horizontal distance from the board when he enters the water?

e) What is Van’s maximum height above the water?

Q15

In a volleyball match, Jenny serves the volleyball at 14 m/s, from a height of 2.5 m above the court. The height of the ball in flight can be estimated using the equation \displaystyle h = -4.9t^2+14t+2.5 , where h is the height, in metres, and t is the time, in seconds, after she serves the ball.

a) What is the maximum height of the volleyball above the court? When does this occur? Round answer to the nearest to the nearest tenth.

b) If a player on the other team contacts the ball at a height of 0.5 In above the court, how long does it take for the ball to reach her? Round to the nearest second.

Q16

The hypotenuse of a right triangle has length 17 cm. The sum of the lengths of the legs is 23 cm. What are their lengths?

Q17

An open-topped box is to be constructed from a square piece of cardboard by removing a square with side length 8 cm from each corner and folding up the edges. The resulting box is to have a volume of 512 cm3. Find the dimensions of the original piece of cardboard.

Q18

Solve the quadratic equation x^2 - 2x - 15 = 0 by factoring and by graphing. Use the graph to find the minimum value of the quadratic relation y = x^2 - 2x - 15.

Q19a

Solve the quadratic equation x^2 + 2x - 15 = 0 by completing the square and by using a computer algebra system.

For what value of b will the quadratic equation 9x^2 + bx + 25 = 0 have one roots? What is the value of the root?
equation x^2 + bx + 10 = 0 have no real roots?