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Completing the Square

Completing the Square 1.12 - Solution

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To rewrite the given quadratic function in vertex form, we will start by calculating (b2)2\left(\frac{b}{2}\right)^2. y=x24x1 y=x^2{\color{#0000FF}{-4}}x-1
(b2)2\left( \dfrac{b}{2} \right)^2
(-42)2\left( \dfrac{{\color{#0000FF}{\text{-}4}}}{2} \right)^2
(-2)2(\text{-}2)^2
222^2
Let's now factor the function by completing the square. We will first add (b2)2=22.\left( \frac{b}{2} \right)^2={\color{#FF0000}{2^2}}. Note that we also have to subtract 22{\color{#FF0000}{2^2}} to leave the function unchanged.
y=x24x1y=x^2-4x-1
y=x24x+22122y=x^2-4x+{\color{#FF0000}{2^2}}-1-{\color{#FF0000}{2^2}}
y=(x2)2122y=(x-2)^2-1-2^2
y=(x2)214y=(x-2)^2-1-4
y=(x2)25y=(x-2)^2-5
Now we know that the vertex form is y=(x2)2+(-5),y=(x-{\color{#009600}{2}})^2+(\textcolor{darkorange}{\text{-}5}), and the vertex of the parabola is (2,-5).({\color{#009600}{2}},\textcolor{orange}{\text{-}5}).