# How do you solve for y in 6y^2-5y=6?

May 25, 2018

color(blue)(y=3/2, -2/3

#### Explanation:

$6 {y}^{2} - 5 y = 6$

$6 {y}^{2} - 5 y - 6 = 0$

$6 {y}^{2} - 9 y + 4 y - 6 = 0$

$3 y \left(2 y - 3\right) + 2 \left(2 y - 3\right) = 0$

$\left(2 y - 3\right) \left(3 y + 2\right) = 0$

$\therefore 2 y - 3 = 0 \mathmr{and} 3 y + 2 = 0$

$t h u s , y = \frac{3}{2} , - \frac{2}{3}$

Plug in color(teal)(y=3/2 and color(teal)(y=-2/3 in the equation.

$6 {\left(\frac{3}{2}\right)}^{2} - 5 \left(\frac{3}{2}\right) = 6 \cdot \frac{9}{4} - \frac{15}{2} = \frac{27}{2} - \frac{15}{2} = \frac{12}{2} = 6$
That proves that color(teal)(y=3/2 is correct

Next, check for $- \frac{2}{3} :$
$6 {\left(- \frac{2}{3}\right)}^{2} - 5 \left(- \frac{2}{3}\right) = 6 \cdot \frac{4}{9} + \frac{10}{3} = \frac{8}{3} + \frac{10}{3} = \frac{18}{3} = 6$

color(blue)(y=3/2, -2/3

May 25, 2018

$y = \frac{2}{3}$
$y = - \frac{3}{2}$

#### Explanation:

Given -

$6 {y}^{2} - 5 y = 6$
divide both sides by 6

${y}^{2} - \frac{5}{6} y = 1$

${y}^{2} - \frac{5}{6} y + \frac{25}{144} = 1 + \frac{25}{144} = \frac{144 + 25}{144} = \frac{169}{144}$

${\left(y + \frac{5}{12}\right)}^{2} = \frac{169}{144}$

$y + \frac{5}{12} = \pm \sqrt{\frac{169}{144}} = \pm \frac{13}{12}$

$y = \frac{13}{12} - \frac{5}{12} = \frac{8}{12} = \frac{2}{3}$

$y = - \frac{13}{12} - \frac{5}{12} = - \frac{18}{12} = - \frac{3}{2}$