# Calculate: 2x^2+3x+2=0

## Expression: $2{x}^{2}+3x+2=0$

Move the constant to the right-hand side and change its sign

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

Divide both sides of the equation by $2$

${x}^{2}+\frac{ 3 }{ 2 }x=-1$

To complete the square, the same value needs to be added to both sides

${x}^{2}+\frac{ 3 }{ 2 }x+?=-1+?$

To complete the square ${x}^{2}+\frac{ 3 }{ 2 }x+\frac{ 9 }{ 16 }={\left( x+\frac{ 3 }{ 4 } \right)}^{2}$ add $\frac{ 9 }{ 16 }$ to the expression

${x}^{2}+\frac{ 3 }{ 2 }x+\frac{ 9 }{ 16 }=-1+?$

Since $\frac{ 9 }{ 16 }$ was added to the left-hand side, also add $\frac{ 9 }{ 16 }$ to the right-hand side

${x}^{2}+\frac{ 3 }{ 2 }x+\frac{ 9 }{ 16 }=-1+\frac{ 9 }{ 16 }$

Use ${a}^{2}+2ab+{b}^{2}={\left( a+b \right)}^{2}$ to factor the expression

${\left( x+\frac{ 3 }{ 4 } \right)}^{2}=-1+\frac{ 9 }{ 16 }$

Calculate the sum

${\left( x+\frac{ 3 }{ 4 } \right)}^{2}=-\frac{ 7 }{ 16 }$

Solve the equation for $x$

$\begin{array} { l }x=-\frac{ 3 }{ 4 }-\frac{ \sqrt{ 7 } }{ 4 }i,\\x=-\frac{ 3 }{ 4 }+\frac{ \sqrt{ 7 } }{ 4 }i\end{array}$

The equation has $2$ solutions

$\begin{array} { l }x_1=-\frac{ 3 }{ 4 }-\frac{ \sqrt{ 7 } }{ 4 }i,& x_2=-\frac{ 3 }{ 4 }+\frac{ \sqrt{ 7 } }{ 4 }i\end{array}$

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