[tex] \huge \boxed{\mathfrak{Question} \downarrow}[/tex]
[tex] \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}[/tex]
[tex] 5x ^{2} - 6x - 3 = 0.[/tex]
Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x²+bx=c.
[tex]5x^{2}-6x-3=0 [/tex]
Add 3 to both sides of the equation.
[tex]5x^{2}-6x-3-\left(-3\right)=-\left(-3\right) [/tex]
Subtracting -3 from itself leaves 0.
[tex]5x^{2}-6x=-\left(-3\right) [/tex]
Subtract -3 from 0.
[tex]5x^{2}-6x=3 [/tex]
Divide both sides by 5.
[tex]\frac{5x^{2}-6x}{5}=\frac{3}{5} \\ [/tex]
Dividing by 5 undoes the multiplication by 5.
[tex]x^{2}+\frac{-6}{5}x=\frac{3}{5} \\ [/tex]
Divide -6 by 5.
[tex]x^{2}-\frac{6}{5}x=\frac{3}{5} \\ [/tex]
Divide -6/5, the coefficient of the x term, by 2 to get -3/5. Then add the square of -3/5 to both sides of the equation. This step makes the left-hand side of the equation a perfect square.
[tex]x^{2}-\frac{6}{5}x+\left(-\frac{3}{5}\right)^{2}=\frac{3}{5}+\left(-\frac{3}{5}\right)^{2} \\ [/tex]
Square -3/5 by squaring both the numerator and the denominator of the fraction.
[tex]x^{2}-\frac{6}{5}x+\frac{9}{25}=\frac{3}{5}+\frac{9}{25} \\ [/tex]
Add 3/5 to 9/25 by finding a common denominator and adding the numerators. Then reduce the fraction to its lowest terms if possible.
[tex]x^{2}-\frac{6}{5}x+\frac{9}{25}=\frac{24}{25} \\ [/tex]
Factor [tex]x^{2}-\frac{6}{5}x+\frac{9}{25}[/tex]. In general, when [tex]x^{2}+bx+c[/tex] is a perfect square, it can always be factored as [tex]\left(x+\frac{b}{2}\right)^{2}[/tex].
[tex]\left(x-\frac{3}{5}\right)^{2}=\frac{24}{25} \\ [/tex]
Take the square root of both sides of the equation.
[tex]\sqrt{\left(x-\frac{3}{5}\right)^{2}}=\sqrt{\frac{24}{25}} \\ [/tex]
Simplify.
[tex]x-\frac{3}{5}=\frac{2\sqrt{6}}{5} \\ x-\frac{3}{5}=-\frac{2\sqrt{6}}{5} [/tex]
Add 3/5 to both sides of the equation.
[tex] \huge\boxed{\boxed{ \bf \: x = \frac{2 \sqrt{6} + 3}{5} \approx 1.58}}\\ \huge \boxed{\boxed{ \bf x=\frac{3-2\sqrt{6}}{5}\approx -0.38}}[/tex]
