# How do you solve 6x^2-21x+15=0?

Mar 26, 2018

x= $\frac{5}{2}$ or $1$

#### Explanation:

Start by simplifying your equation by factoring out a 3:
$3 \left(2 {x}^{2} - 7 x + 5\right) = 0$
$2 {x}^{2} - 7 x + 5 = 0$

This equation cannot be factored with whole numbers, so you should use the quadratic formula:

$\frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$, knowing that $a {x}^{2} + b x + c$

So now:
$\frac{- \left(- 7\right) \pm \sqrt{{\left(- 7\right)}^{2} - 4 \left(2\right) \left(5\right)}}{2 \left(2\right)}$
$\frac{7 \pm \sqrt{49 - 4 \left(2\right) \left(5\right)}}{4}$
$\frac{7 \pm \sqrt{49 - 40}}{4}$
$\frac{7 \pm \sqrt{9}}{4}$
$\frac{7 \pm 3}{4}$
$\frac{10}{4}$ or $\frac{4}{4}$=
$\frac{5}{2}$ or $1$

x= $\frac{5}{2}$ or $1$

Mar 26, 2018

$x = \frac{21}{12} \pm \sqrt{\frac{54}{96}}$

#### Explanation:

In order to complete the square move the last term (term without $x$) to other side of equation
${x}^{2} - \frac{21}{6} x = - \frac{15}{6}$

Then you want to find a piece that allows you to find a square square of the left hand side
i.e. ${a}^{2} + 2 a b + {b}^{2} = {\left(a + b\right)}^{2}$
or
${a}^{2} - 2 a b + {b}^{2} = {\left(a - b\right)}^{2}$

In this equation $x = a$, $2 a b = - \frac{21}{6} x$ so as $x = a$ we know that $2 b = - \frac{21}{6}$ so to complete the square we just need ${b}^{2}$ so if we half and square $2 b$ we will get it so ${b}^{2} = {\left(\frac{21}{12}\right)}^{2}$

So if we add this term to both sides we get

${x}^{2} - \frac{21}{6} x + {\left(\frac{21}{12}\right)}^{2} = - \frac{15}{6} + {\left(\frac{21}{12}\right)}^{2}$
Now the left hand side can be simplifed into merely ${\left(a - b\right)}^{2}$

${\left(x - \frac{21}{12}\right)}^{2} = - \frac{15}{6} + \frac{441}{144}$

${\left(x - \frac{21}{12}\right)}^{2} = - \frac{15}{6} + \frac{49}{16}$

Find a common multiple for 16 and 6 and add them together

${\left(x - \frac{21}{12}\right)}^{2} = - \frac{240}{96} + \frac{294}{96}$

${\left(x - \frac{21}{12}\right)}^{2} = \frac{54}{96}$

Square root both sides

$x - \frac{21}{12} = \pm \sqrt{\frac{54}{96}}$

$x = \frac{21}{12} \pm \sqrt{\frac{54}{96}}$