Solve for: x=3x^2-4x-3

Expression: $x=3{x}^{2}-4x-3$

Move the expression to the left-hand side and change its sign

$x-3{x}^{2}+4x+3=0$

Collect like terms

$5x-3{x}^{2}+3=0$

Use the commutative property to reorder the terms

$-3{x}^{2}+5x+3=0$

Change the signs on both sides of the equation

$3{x}^{2}-5x-3=0$

Identify the coefficients $a$, $b$ and $c$ of the quadratic equation

$\begin{array} { l }a=3,& b=-5,& c=-3\end{array}$

Substitute $a=3$, $b=-5$ and $c=-3$ into the quadratic formula $x=\frac{ -b\pm\sqrt{ {b}^{2}-4ac } }{ 2a }$

$x=\frac{ -\left( -5 \right)\pm\sqrt{ {\left( -5 \right)}^{2}-4 \times 3 \times \left( -3 \right) } }{ 2 \times 3 }$

When there is a $-$ in front of an expression in parentheses, change the sign of each term of the expression and remove the parentheses

$x=\frac{ 5\pm\sqrt{ {\left( -5 \right)}^{2}-4 \times 3 \times \left( -3 \right) } }{ 2 \times 3 }$

Evaluate the power

$x=\frac{ 5\pm\sqrt{ 25-4 \times 3 \times \left( -3 \right) } }{ 2 \times 3 }$

Calculate the product

$x=\frac{ 5\pm\sqrt{ 25+36 } }{ 2 \times 3 }$

Multiply the numbers

$x=\frac{ 5\pm\sqrt{ 25+36 } }{ 6 }$

Add the numbers

$x=\frac{ 5\pm\sqrt{ 61 } }{ 6 }$

Write the solutions, one with a $+$ sign and one with a $-$ sign

$\begin{array} { l }x=\frac{ 5+\sqrt{ 61 } }{ 6 },\\x=\frac{ 5-\sqrt{ 61 } }{ 6 }\end{array}$

The equation has $2$ solutions

$\begin{align*}&\begin{array} { l }x_1=\frac{ 5-\sqrt{ 61 } }{ 6 },& x_2=\frac{ 5+\sqrt{ 61 } }{ 6 }\end{array} \\&\begin{array} { l }x_1\approx-0.468375,& x_2\approx2.13504\end{array}\end{align*}$