# How do you solve 6-7x=3x²?

Jun 22, 2017

$x = \frac{2}{3} , x = - 3$

#### Explanation:

Put all terms onto one side of the equation so that everything is equal to $0$.
$6 - 7 x = 3 {x}^{2}$
$6 = 3 {x}^{2} + 7 x$
$0 = 3 {x}^{2} + 7 x - 6$

Factor.
$3 {x}^{2} + 7 x - 6 = 0$
$\left(3 x - 2\right) \left(x + 3\right) = 0$

Set each factor equal to $0$. Make sure to do this for both.
$3 x - 2 = 0$
$3 x = 2$
$x = \frac{2}{3}$

$x + 3 = 0$
$x = - 3$

Answers: $x = \frac{2}{3} , x = - 3$

Jun 22, 2017

$\frac{2}{3} \mathmr{and} - 3$

#### Explanation:

$y = 3 {x}^{2} + 7 x - 6 = 0$

Use the new transforming method (Socratic Search)
Transformed equation:

$y ' = {x}^{2} + 7 x - 18 = 0 \text{ } \rightarrow \left(a \times c = 3 \left(- 6\right) = - 18\right)$

Proceed. Find 2 real roots of y', then, divide them by $a = 3.$

Find $2$ numbers knowing the sum (-b = -7))
and the product $\left(a \times c = - 18\right)$.

They are: $2 , \mathmr{and} \left(- 9\right) .$

Consequently, the $2$ real roots of $y$ are:

${x}_{1} = \frac{2}{a} = \frac{2}{3}$, and ${x}_{2} = - \frac{9}{3} = - 3$

Note. This new method avoids the lengthy factoring by grouping and solving the 2 binomials.