Completing the Square (Edexcel GCSE Maths)

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FrontCompleting the Square
When completing the square for the expression $x^{2} + b x + c$ , explain how to find the value of $p$ in the expression ${(x + p)}^{2} + q$ .

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Cards in this collection (5)

When completing the square for the expression $x^{2} + b x + c$ , explain how to find the value of $p$ in the expression ${(x + p)}^{2} + q$ .
Completing the square for $x^{2} + b x + c$ gives the form ${(x + p)}^{2} + q$ .
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 $y = {(x + 3)}^{2} + 5$ are $(3, 5)$ .
False.
The coordinates of the turning point (vertex) of the quadratic curve $y = {(x + 3)}^{2} + 5$ are not $(3, 5)$ .
The correct coordinates are $(- 3, 5)$ .
This is a common mistake in the exam! If $y = {(x + p)}^{2} + q$ is the curve, then $(- p, q)$ are the coordinates of the turning point.
What is the first step to completing the square of the quadratic expression $a x^{2} + b x + c$ where $a$ is not equal to 1, e.g. $3 x^{2} + 12 x - 8$ ?
The first step to completing the square of the quadratic expression $a x^{2} + b x + c$ where $a$ is not equal to 1 is to factorise out $a$ .
This gives $a [x^{2} + \frac{b}{a} x + \frac{c}{a}]$ .
You can then complete the square inside the big brackets.
E.g. to complete the square of the expression $3 x^{2} + 12 x - 8$ , the first step would be to factorise out $3$ , giving $3 [x^{2} + 4 x - \frac{8}{3}]$ .
True or False?
The coordinates of the turning points on the curves $y = {(x + 2)}^{2} + 4$ and $y = 3 {(x + 2)}^{2} + 4$ are the same.
True.
The coordinates of the turning points on the curves $y = {(x + 2)}^{2} + 4$ and $y = 3 {(x + 2)}^{2} + 4$ are the same.
The coordinates of the turning point on $y = a {(x + p)}^{2} + q$ are always $(- p, q)$ , regardless of the value of $a$ (even if $a < 0$ ).
Explain how writing $x^{2} - 4 x + 9$ in the form ${(x - 2)}^{2} + 5$ shows that any output of the function $f (x) = x^{2} - 4 x + 9$ is always greater than or equal to 5.
If $x^{2} - 4 x + 9$ can be written as ${(x - 2)}^{2} + 5$ by completing the square, then $f (x) = x^{2} - 4 x + 9$ can be written as $f (x) = {(x - 2)}^{2} + 5$ .
The coordinates of the turning point will be $(2, 5)$ .
As this is a positive quadratic, the turning point will be a minimum and the therefore the value of $f (x)$ will always be greater than or equal to 5.

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