Nathanial Frost

2023-02-18

How to write $f\left(x\right)={x}^{2}+4x+1$ in vertex form?

Rhys Murphy

The equation in vertex form is $f\left(x\right)=a{\left(x-h\right)}^{2}+k$
Here, we have $f\left(x\right)={x}^{2}+4x+1$, so we have $a=1$
So for converting to this form, we complete the square using ${\left(x+a\right)}^{2}={x}^{2}+2ax+{a}^{2}$ for which we have to identify $a$ and then add and subtract ${a}^{2}$ Thus,
$f\left(x\right)=\underline{{x}^{2}+2×2×x+{2}^{2}}-{2}^{2}+1$
$={\left(x+2\right)}^{2}-4+1$
$={\left(x-\left(-2\right)\right)}^{2}-3$

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