AS Mathematics

1
AS Mathematics

• Algebra Solving quadratic equations by
  completing the square

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Objectives

• Be able to recognise and factorise perfect
  squares
• Be able to solve equations by completing the
  square

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Solving equations by completing the square
This method for solving quadratic equations
involves manipulating the equation so that it
includes a perfect square
(a b)2 a2 2ab b2
Lets look at some perfect squares first!
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Perfect squares
(x 1)2
x2 2x 1
(x 2)2
x2 4x 4
x2 6x 9
(x 3)2
(x 4)2
x2 8x 16
x2 10x 25
(x 5)2
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Example 1
Solve the equation x2 6x 2
Halve the coefficient of x ? (x 3)2
BUT (x 3)2 x2 6x 9
Subtract 32 i.e. 9 to get x2 6x
6
So x2 6x (x 3)2 - 9
Which gives (x 3)2 9 2
(x 3)2 2 9
(x 3)2 11
x 3 v11
x -3 v11
x -6.32 or x 0.32 (2 d.p.)
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Example 2
Solve the equation x2 - 8x - 6 0
x2 - 8x 6
Halve the coefficient of x ? (x - 4)2
Subtract 42
(x - 4)2 16 6
(x - 4)2 6 16
(x - 4)2 22
x -0.69 or x 8.69 (2 d.p.)
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How can we find the equation of the line of
symmetry of an equation using the completed
square form?
(see example 1)
Look at y x2 6x -2
y (x 3)2 - 9 - 2
y (x 3)2 - 11
You need a graphical calculator or a graphing
package!!
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Start by drawing y x2
y x2
Line of symmetry at x 0
10
Translate to get 3 places to the left to get
y (x 3)2
y (x 3)2
y x2
Line of symmetry at x -3
11
Translate 11 places down to get y (x 3)2 -
11
y (x 3)2
Line of symmetry at x -3
y x2
y (x 3)2 -11
Minimum point at (-3, -11)
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• Repeat this method to find
• the line of symmetry and
• ii) the minimum point
• of y x2 8x 6