Exercises


Lesson 1: Expanding and Simplifying


  1. Expand and simplify the following expressions.
    1. \(-4x(x+5)\)
    2. \((3x+4)(2x-7)\)
    3. \((2x+1)^2\)
    4. \(\dfrac12(x-6)(x-8)\)
  2. Write each pair of quadratic relations in standard form.  Which of the pairs represent the same relation?
    1. \(y=-2(x-1)^2+8\) and \(y=-2(x-3)(x+1)\)
    2. \(y=\dfrac13(x+3)^2-12\) and \(y=\dfrac13(x-3)(x+6)\)
  3. Write the equation, in standard form, of the quadratic relation with:
    1. Zeros of \(2\) and \(6\), and a \(y\)-intercept of \(3\).
    2. Vertex \((-5, 0)\), passing through the point \((5, -300)\).

    3. A minimum of \(-3\) and the following graph:

      A parabola that opens upwards with points (negative 6, 2) and (4, 2) marked.

    1. Let three consecutive positive integers be \(x-1\), \(x\), and \(x+1\). Show that the square of the smallest integer subtracted from the square of the largest integer equals four times the middle integer.  
    2. Using the fact found in part a), evaluate the following.
      1. \(251^2-249^2\)
      2. \(2001^2-1999^2\)
  5. Expand and simplify the following expressions.
    1. \(3x(x+1)(x-2)\)
    2. \((2x+1)(x-1)(3x+2)\)
  6. A spherical balloon has a volume of \(V=\dfrac43 \pi r^3\).  
    1. Write a simplified expression for the difference in volume, if the radius of the balloon decreases by \(3\) units.  Let \(\Delta V\) represent the larger volume minus the smaller volume, and let \(r\) represent the larger radius. 
    2. A balloon has a radius of \(8\) units.  The radius then decreases by \(3\) units. What is \(\Delta V\) in this case? (Provide an exact answer.)
    1. Expand and simplify \((x+1)^4\).  Hint:  Write the expression as \((x+1)^4=(x+1)^2(x+1)^2\). 
    2. How many terms will be in the expansion of \((x+1)^5\)?

Lesson 2: Factoring — Common and Trinomials


  1. Determine the zeros.
    1. \(y=3x^2+3x\)
    2. \(y=4x^2+4x+1\)
    3. \(y=3x^2+\dfrac{1}{2}x-\dfrac{1}{2}\) (Hint: Common factor \(\dfrac12\) out of the right side).
  2. Factor completely.
    1. \(7x^2+42x-49\)
    2. \(5x^3y-15x^2y-140xy\)
    3. \(6x^2-xy-15y^2\)
    1. Factor \(a^2+2ab+b^2\).
    2. Use your answer to part a) to factor the following.
      1. \(x^2+14x+49\)
      2. \(9x^2+12x+4\)
  4. The polynomial \(x^2+kx+16\) is factorable over the integers. Determine all possible integer values of \(k\).
  5. Determine the factored form equation and the vertex of the parabola with \(x\)-intercepts \(-7\) and \(9\) and \(y\)-intercept \(63\).
  6. The quadratic relation \(y=2x^2+bx-5\) has a zero at \(x=-5\).
    1. Determine the value of \(b\).
    2. Write the relation in factored form.
  7. A projectile is fired from a cannon and follows a parabolic (quadratic) path. Its height, \(h\) in metres, is given by the equation \(h=-4.9t^2+bt\) where \(t\) is the time in seconds. The projectile is in the air for exactly \(12\) seconds.
    1. Determine the value of \(b\).
    2. Determine the maximum height of the projectile.
  8. The product of two consecutive integers is \(132\). Determine the integers.
  9. A card is \(3\) cm longer than it is wide. The area of the card is \(54\) square cm. Determine the dimensions of the card.

Lesson 3: Factoring — Difference of Squares and Perfect Squares


  1. Factor fully.
    1. \(x^2-9\)
    2. \(16x^2-25\)
    3. \(81-x^2\)
    4. \(12x^3-3xy^2\)
    5. \(5x^8-405\)
    1. Using integer coefficients, determine two possible standard form quadratic relations that have zeros \(x=\pm \dfrac{7}{3}\). 
    2. Determine all possible standard form quadratic relations that have zeros \(x=\pm \dfrac{q}{p}\) for all real numbers \(\)\(p\) and \(q\) with \(p \ne 0\).
  3. Factor fully.
    1. \(x^2-8x+16\)
    2. \(16x^2+24x+9\)
    3. \(2x^3+40x^2+200x\)
    4. \(3x^3y-24x^2y+48xy\)
  4. Determine the missing terms, given that each of the following expressions is a perfect square. Write the factored form of each expression.
    1. \(\boxed{\phantom{\square\square}}-12xy+9y^2\)
    2. \(49x^6+42x^3y^5 +\boxed{\phantom{\square\square}}\)
    3. \(100x^6-300x^3+\boxed{\phantom{\square\square}}\)
  5. Complete the table.
    Relation Factored Form Zero(s) Axis of Symmetry Vertex
    \(y=x^2-9\)        
    \(y=x^2-12x+36\)        
    \(y=9x^2+12x+4\)        
  6. Factor fully by grouping.
    1. \(10x^2+2xy-15x-3y\)
    2. \(3x^2+2xy-9x-6y\)
    3. \(15xy+1+3x+5y\)
  7. Use substitution to factor fully.
    1. \((x^2+2x)^2-2(x^2+2x)-3\)
    2. \((y^4+y^2)^2-92(y^4+y^2)+180\)
  8. The relation \(y=(x^2+2x)^2-2(x^2+2x)-3\) is not quadratic, yet it is still possible to determine the zeros. Determine the zeros of \(y=(x^2+2x)^2-2(x^2+2x)-3\).
    1. Using integer coefficients, determine two possible standard form quadratic relations that have exactly one zero, namely \(x=-\dfrac{3}{4}\).
    2. Determine all possible standard form quadratic relations that have exactly one zero, namely \(x=\dfrac{q}{p}\), for all real numbers \(\)\(p\) and \(q\) with \(p \ne 0\).
  10. One factor of \(x^3+3x^2-4\) is \(x-1\). Factor \(x^3+3x^2-4\) fully.

Lesson 4: Completing the Square


  1. Determine the value of \(c\) that will make each of the following trinomials a perfect square.
    1. \(x^2+10x+c\)
    2. \(x^2-200x+c\)
    3. \(x^2+5x+c\)
    4. \(x^2-\dfrac12 x +c\)
  2. Write the following in vertex form by completing the square.
    1. \(y=x^2+8x+1\)
    2. \(y=2x^2+12x\)
    3. \(y=-x^2-4x+9\)
    4. \(y=5x^2-50x+132\)
    5. \(y=x^2-3x+9\)
    6. \(y=\dfrac34 x^2-9x+5\)
  3. Complete the chart.
    Quadratic Vertex Form Axis of Symmetry Vertex
    \(y=x^2-9\)      
    \(y=x^2-12x+36\)      
    \(y=9x^2+18x+4\)      
  4. Change each of the following vertex form equations to factored form.
    1. \(y=(x+3)^2-16\)
    2. \(y=-3 (x-1)^2+12\)
  5. For the relation \(y=(2x-5)(x+3)\),
    1. find the zeros.
    2. find the \(x\)-coordinate of the vertex.
    3. find the \(y\)-coordinate of the vertex.
    4. write the equation in vertex form using your answers in parts b) and c).
  6. At a Canada Day fireworks show, the height of one of the rockets after it is launched is given by the equation \(h=-4.9t^2+78.4t+10\), where \(h\) is in metres and \(t\) is the time, in seconds, after the launch.

    The rocket explodes when it reaches its maximum height above the ground.

    1. Complete the square on this quadratic relation to find the time when the rocket will explode.
    2. How many metres above the ground is the rocket at that time?
  7. A lifeguard has used \(620\) m of marker buoys to rope off a safe swimming area. One side of the swimming area is adjacent to the beach.
    1. Write an equation in standard form relating the area and the length of the swimming area (i.e., the length of the swimming area that is parallel to the beach).
    2. Complete the square to write the relation in vertex form.
    3. Identify the maximum area and the dimensions that produce it.
    1. Complete the square on the general form \(y=ax^2+bx+c\) to create the vertex form: \(y=a(x-h)^2+k\).
    2. What is the relationship between the value of \(h\) and the values of \(a\) and \(b\)?
    3. Check your answers to Exercises 2b and 2d using this relationship.