# How do you solve 5x^2-30x+5=0 by completing the square?

Aug 4, 2016

$x = 3 - 2 \sqrt{2}$ or $x = 3 + 2 \sqrt{2}$

#### Explanation:

To solve $5 {x}^{2} - 30 x + 5 = 0$, let us first divide both sides by $5$ whic gives us

${x}^{2} - 6 x + 1 = 0$

Now to complete square for ${x}^{2} - 6 x$, recalling identity ${\left(x - a\right)}^{2} = {x}^{2} - 2 a x + {a}^{2}$, we must add and subtract square of half of the coefficient of $x$ i.e. ${\left(\frac{6}{2}\right)}^{2} = 9$, hence we have

${x}^{2} - 6 x + 9 - 9 + 1 = 0$ or

${\left(x - 3\right)}^{2} - 8 = 0$ or ${\left(x - 3\right)}^{2} - {\left(2 \sqrt{2}\right)}^{2} = 0$ which can be factorized as

$\left(x - 3 + 2 \sqrt{2}\right) \left(x - 3 - 2 \sqrt{2}\right) = 0$

Hence $x = 3 - 2 \sqrt{2}$ or $x = 3 + 2 \sqrt{2}$