New Notes Blog: Factor by completing the square for` f(x)= x^2 -4x +9`

Saturday, 13 June 2015

Factor by completing the square for $\displaystyle{f{{\left({x}\right)}}}={{x}}^{{2}}-{4}{x}+{9}$

Express $\displaystyle{f{{\left({x}\right)}}}={{x}}^{{2}}-{4}{x}+{9}$ as $\displaystyle{{x}}^{{2}}-{2}\cdot{x}\cdot{2}+{{2}}^{{2}}-{{2}}^{{2}}+{9}={{\left({x}-{2}\right)}}^{{2}}-{4}+{9}={{\left({x}-{2}\right)}}^{{2}}+{5}.$ We used the formula $\displaystyle{{\left({a}-{b}\right)}}^{{2}}={{a}}^{{2}}-{2}{a}{b}+{{b}}^{{2}}$ in the reverse direction.

We see that this function is always positive for real $\displaystyle{x},$ therefore it cannot be factored using real coefficients.

But it can using complex numbers: $\displaystyle-{5}={{\left({i}\sqrt{{{5}}}\right)}}^{{2}}$ and

$\displaystyle{f{{\left({x}\right)}}}={{\left({x}-{2}\right)}}^{{2}}+{5}={{\left({x}-{2}\right)}}^{{2}}-{{\left({i}\sqrt{{{5}}}\right)}}^{{2}}={\left({x}-{2}-{i}\sqrt{{{5}}}\right)}{\left({x}-{2}+{i}\sqrt{{{5}}}\right)}.$

Here we used the formula $\displaystyle{{a}}^{{2}}-{{b}}^{{2}}={\left({a}-{b}\right)}{\left({a}+{b}\right)}.$

The answer is impossible for real coefficients and $\displaystyle{\left({x}-{2}-{i}\sqrt{{{5}}}\right)}{\left({x}-{2}+{i}\sqrt{{{5}}}\right)}$ for complex coefficients.

New Notes Blog

Saturday, 13 June 2015

Factor by completing the square for $\displaystyle{f{{\left({x}\right)}}}={{x}}^{{2}}-{4}{x}+{9}$

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In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?

Saturday, 13 June 2015

Factor by completing the square for

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In &quot;By the Waters of Babylon,&quot; under the leadership of John, what do you think the Hill People will do with their society?

Factor by completing the square for $\displaystyle{f{{\left({x}\right)}}}={{x}}^{{2}}-{4}{x}+{9}$

In "By the Waters of Babylon," under the leadership of John, what do you think the Hill People will do with their society?