How do you solve using the completing the square method x^2+10x-2=0?

May 28, 2016

The solutions are:
color(green)(x = 3sqrt 3 - 5 , color(green)(x = -3sqrt 3 -5

Explanation:

${x}^{2} + 10 x - 2 = 0$

${x}^{2} + 10 x = 2$

To write the Left Hand Side as a Perfect Square, we add 25 to both sides:

${x}^{2} + 10 x + \textcolor{b l u e}{25} = 2 + \textcolor{b l u e}{25}$

${x}^{2} + 2 \cdot x \cdot 5 + {5}^{2} = 27$

Using the Identity color(blue)((a+b)^2 = a^2 + 2ab + b^2, we get

${\left(x + 5\right)}^{2} = 27$

$x + 5 = \sqrt{27}$ or $x + 5 = - \sqrt{27}$

(Note: prime factorising 27; 27 = 3 * 3 * 3 = 3^3

So, $\sqrt{27} = \sqrt{{3}^{3}} = 3 \sqrt{3}$ )

$x + 5 = 3 \sqrt{3}$ or $x + 5 = - 3 \sqrt{3}$

color(green)(x = 3sqrt 3 - 5 , color(green)(x = -3sqrt 3 -5 #