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Completing the Square (Edexcel IGCSE Maths A)

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  • FrontCompleting the Square

    When completing the square for the expression x squared plus b x plus c, explain how to find the value of p in the expression open parentheses x plus p close parentheses squared plus q.

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  • When completing the square for the expression x squared plus b x plus c, explain how to find the value of p in the expression open parentheses x plus p close parentheses squared plus q.

    Completing the square for x squared plus b x plus c gives the form open parentheses x plus p close parentheses squared plus q.

    The value of p is half of the value of b.

  • True or False?

    The coordinates of the turning point (vertex) of the quadratic curve y equals open parentheses x plus 3 close parentheses squared plus 5 are open parentheses 3 comma space 5 close parentheses.

    False.

    The coordinates of the turning point (vertex) of the quadratic curve y equals open parentheses x plus 3 close parentheses squared plus 5 are not open parentheses 3 comma space 5 close parentheses.

    The correct coordinates are open parentheses negative 3 comma space 5 close parentheses.

    This is a common mistake in the exam! If y equals open parentheses x plus p close parentheses squared plus q is the curve, then open parentheses negative p comma space q close parentheses are the coordinates of the turning point.

  • What is the first step to completing the square of the quadratic expression a x squared plus b x plus c where a is not equal to 1, e.g. 3 x squared plus 12 x minus 8?

    The first step to completing the square of the quadratic expression a x squared plus b x plus c where a is not equal to 1 is to factorise out a.

    This gives a open square brackets x squared plus b over a x plus c over a close square brackets.

    You can then complete the square inside the big brackets.

    E.g. to complete the square of the expression 3 x squared plus 12 x minus 8, the first step would be to factorise out 3, giving 3 open square brackets x squared plus 4 x minus 8 over 3 close square brackets.

  • True or False?

    The coordinates of the turning points on the curves y equals open parentheses x plus 2 close parentheses squared plus 4 and y equals 3 open parentheses x plus 2 close parentheses squared plus 4 are the same.

    True.

    The coordinates of the turning points on the curves y equals open parentheses x plus 2 close parentheses squared plus 4 and y equals 3 open parentheses x plus 2 close parentheses squared plus 4 are the same.

    The coordinates of the turning point on y equals a open parentheses x plus p close parentheses squared plus q are always open parentheses negative p comma space q close parentheses, regardless of the value of a (even if a less than 0).

  • Explain how writing x squared minus 4 x plus 9 in the form open parentheses x minus 2 close parentheses squared plus 5 shows that any output of the function straight f open parentheses x close parentheses equals x squared minus 4 x plus 9 is always greater than or equal to 5.

    If x squared minus 4 x plus 9 can be written as open parentheses x minus 2 close parentheses squared plus 5 by completing the square, then straight f open parentheses x close parentheses equals x squared minus 4 x plus 9 can be written as straight f open parentheses x close parentheses equals open parentheses x minus 2 close parentheses squared plus 5.

    The coordinates of the turning point will be open parentheses 2 comma space 5 close parentheses.

    As this is a positive quadratic, the turning point will be a minimum and the therefore the value of straight f open parentheses x close parentheses will always be greater than or equal to 5.

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