Transcript OCF.01.3 - Completing the Square Technique

OCF.01.3 - Completing the
Square Technique
MCR3U - Santowski
(A) Perfect Square Trinomials
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Polynomials like (x + 4)2 are called perfect squares.
When expanded, (x + 4)2 equals x2 + 8x + 16.
When x2 + 8x + 16 is factored, it becomes (x + 4)2
Notice the relationship between the x term (8) and the constant
term (16) ?????
If not, try another perfect square, say (x + 6)2 and x2 + 12x + 36
(12 and 36??)
ex. Find the value of b that makes each expression a perfect square
(a) x2 + 4x + b
(b) x2 + 10x + b
(c) x2 + 14x + b
(d) x2 - 12x + b
(e) x2 – 5x + b
(B) The Vertex Form of a Quadratic Equation
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The quadratic equation y = a(x – h)2 + k is the vertex form and
conveniently gives us the location of the vertex of the parabola,
which is at (h,k).
In many cases the equation of a quadratic may be presented in
standard form y = ax2 + bx + c
We have to go through the “completing the square” technique in
order to convert the equation to the vertex form.
Then, we can identify the vertex and if the vertex represents a
maximum (parabola opens down) or a minimum (parabola opens
up) by the sign of the a term. If a is positive, the parabola opens up
and the vertex represents a min. point and if a is negative, the
parabola opens down and the vertex represents a max. point.
(C) The Completing the Square Method
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Steps Involved in the technique:
Example using y = 2x2 + 12x - 3
(i) Factor the leading co-efficient of 2 (the value of a) from both the x2 and
x terms  y = 2(x2 + 6x) - 3
(ii) Find the constant term that must be present in order to create a perfect
square trinomial  y = 2(x2 + 6x + 9 ) - 3  BUT *****?????
(iii) Make an adjustment in the equation because of the extra value added
(+9)  y = 2(x2 + 6x + 9 - 9) - 3
(iv) Group the perfect square trinomial. Move the subtracted value out of
the bracket  y = 2(x2 + 6x + 9) – 2x9 - 3
(v) Factor the perfect square and collect like terms  y = 2(x + 3)2 - 21
(D) Further Examples of the Method
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ex
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y = x2 + 8x + 15
y = -2x2 + 12x - 7
y = -0.3f2 - 2.4f + 7.3
y = 5x – 3x2
In each example, identify the vertex and the direction of opening of
the parabola
Internet Links
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Completing the Square by James Brennan
Completing the Square: Solving Quadratics from
Purple Math
Completing the Square from Bethany Lutheran
College
(E) Homework
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Nelson Text, p306, Q1abcdefg, 2abcdef, 3abc, 5abcd, 7, 9