Question
Solve the following quadratic equation by completing the square method.
$m^2-5 m=-3$
$m^2-5 m=-3$
✓
Answer
$m ^2-5 m+3=0 \Rightarrow m ^2-5 m+\frac{25}{4}-\frac{25}{4}+3=0$ (Adding and Subtracting $\frac{25}{4}$)
$ \Rightarrow\left( m ^2-5 m+\frac{25}{4}\right)=\frac{25}{4}-3$
$\Rightarrow\left( m -\frac{5}{2}\right)^2=\frac{25-12}{4} $
$ \Rightarrow\left( m -\frac{5}{2}\right)^2=\frac{13}{4} $
$ \Rightarrow m -\frac{5}{2}=\sqrt{\frac{13}{4}} $
$ \Rightarrow m -\frac{5}{2}= \pm \frac{\sqrt{13}}{2} $
$ \Rightarrow m -\frac{5}{2}=\frac{\sqrt{13}}{2} \text { or } m -\frac{5}{2}=-\frac{\sqrt{13}}{2} $
$ \Rightarrow m =\frac{\sqrt{13}}{2}+\frac{5}{2} \text { or } m =-\frac{\sqrt{13}}{2}-\frac{5}{2} $
$ \Rightarrow m =\frac{\sqrt{13}+5}{2} \text { or } m =\frac{-\sqrt{13}-5}{2}$
$ \Rightarrow\left( m ^2-5 m+\frac{25}{4}\right)=\frac{25}{4}-3$
$\Rightarrow\left( m -\frac{5}{2}\right)^2=\frac{25-12}{4} $
$ \Rightarrow\left( m -\frac{5}{2}\right)^2=\frac{13}{4} $
$ \Rightarrow m -\frac{5}{2}=\sqrt{\frac{13}{4}} $
$ \Rightarrow m -\frac{5}{2}= \pm \frac{\sqrt{13}}{2} $
$ \Rightarrow m -\frac{5}{2}=\frac{\sqrt{13}}{2} \text { or } m -\frac{5}{2}=-\frac{\sqrt{13}}{2} $
$ \Rightarrow m =\frac{\sqrt{13}}{2}+\frac{5}{2} \text { or } m =-\frac{\sqrt{13}}{2}-\frac{5}{2} $
$ \Rightarrow m =\frac{\sqrt{13}+5}{2} \text { or } m =\frac{-\sqrt{13}-5}{2}$
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