$\color{#FF6800}{ 40 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } = 75$
$ $ Organize the expression $ $
$\color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 40 } \color{#FF6800}{ x } = 75$
$- 5 x ^ { 2 } + 40 x = \color{#FF6800}{ 75 }$
$ $ Move the expression to the left side and change the symbol $ $
$- 5 x ^ { 2 } + 40 x \color{#FF6800}{ - } \color{#FF6800}{ 75 } = 0$
$\color{#FF6800}{ - } \color{#FF6800}{ 5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ + } \color{#FF6800}{ 40 } \color{#FF6800}{ x } \color{#FF6800}{ - } \color{#FF6800}{ 75 } = \color{#FF6800}{ 0 }$
$ $ Change the symbols of both sides of the equation $ $
$\color{#FF6800}{ 5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 40 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 75 } = \color{#FF6800}{ 0 }$
$\color{#FF6800}{ 5 } \color{#FF6800}{ x } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 40 } \color{#FF6800}{ x } \color{#FF6800}{ + } \color{#FF6800}{ 75 } = \color{#FF6800}{ 0 }$
$ $ Solve the quadratic equation $ ax^{2}+bx+c=0 $ using the quadratic formula $ \dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } \color{#FF6800}{ 40 } \right ) \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 40 } \right ) ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } \color{#FF6800}{ \times } \color{#FF6800}{ 75 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } }$
$x = \dfrac { \color{#FF6800}{ - } \left ( \color{#FF6800}{ - } 40 \right ) \pm \sqrt{ \left ( - 40 \right ) ^ { 2 } - 4 \times 5 \times 75 } } { 2 \times 5 }$
$ $ Simplify Minus $ $
$x = \dfrac { 40 \pm \sqrt{ \left ( - 40 \right ) ^ { 2 } - 4 \times 5 \times 75 } } { 2 \times 5 }$
$x = \dfrac { 40 \pm \sqrt{ \left ( \color{#FF6800}{ - } \color{#FF6800}{ 40 } \right ) ^ { \color{#FF6800}{ 2 } } - 4 \times 5 \times 75 } } { 2 \times 5 }$
$ $ Remove negative signs because negative numbers raised to even powers are positive $ $
$x = \dfrac { 40 \pm \sqrt{ 40 ^ { 2 } - 4 \times 5 \times 75 } } { 2 \times 5 }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 40 } \pm \sqrt{ \color{#FF6800}{ 40 } ^ { \color{#FF6800}{ 2 } } \color{#FF6800}{ - } \color{#FF6800}{ 4 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } \color{#FF6800}{ \times } \color{#FF6800}{ 75 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } }$
$ $ Organize the expression $ $
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 40 } \pm \sqrt{ \color{#FF6800}{ 100 } } } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } } }$
$x = \dfrac { 40 \pm \sqrt{ \color{#FF6800}{ 100 } } } { 2 \times 5 }$
$ $ Organize the part that can be taken out of the radical sign inside the square root symbol $ $
$x = \dfrac { 40 \pm \color{#FF6800}{ 10 } } { 2 \times 5 }$
$x = \dfrac { 40 \pm 10 } { \color{#FF6800}{ 2 } \color{#FF6800}{ \times } \color{#FF6800}{ 5 } }$
$ $ Multiply $ 2 $ and $ 5$
$x = \dfrac { 40 \pm 10 } { \color{#FF6800}{ 10 } }$
$\color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 40 } \pm \color{#FF6800}{ 10 } } { \color{#FF6800}{ 10 } } }$
$ $ Separate the answer $ $
$\begin{array} {l} \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 40 } \color{#FF6800}{ + } \color{#FF6800}{ 10 } } { \color{#FF6800}{ 10 } } } \\ \color{#FF6800}{ x } = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 40 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } } { \color{#FF6800}{ 10 } } } \end{array}$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 40 } \color{#FF6800}{ + } \color{#FF6800}{ 10 } } { 10 } \\ x = \dfrac { 40 - 10 } { 10 } \end{array}$
$ $ Add $ 40 $ and $ 10$
$\begin{array} {l} x = \dfrac { \color{#FF6800}{ 50 } } { 10 } \\ x = \dfrac { 40 - 10 } { 10 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 50 } } { \color{#FF6800}{ 10 } } } \\ x = \dfrac { 40 - 10 } { 10 } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 5 } } { \color{#FF6800}{ 1 } } } \\ x = \dfrac { 40 - 10 } { 10 } \end{array}$
$\begin{array} {l} x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 5 } } { \color{#FF6800}{ 1 } } } \\ x = \dfrac { 40 - 10 } { 10 } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = \color{#FF6800}{ 5 } \\ x = \dfrac { 40 - 10 } { 10 } \end{array}$
$\begin{array} {l} x = 5 \\ x = \dfrac { \color{#FF6800}{ 40 } \color{#FF6800}{ - } \color{#FF6800}{ 10 } } { 10 } \end{array}$
$ $ Subtract $ 10 $ from $ 40$
$\begin{array} {l} x = 5 \\ x = \dfrac { \color{#FF6800}{ 30 } } { 10 } \end{array}$
$\begin{array} {l} x = 5 \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 30 } } { \color{#FF6800}{ 10 } } } \end{array}$
$ $ Do the reduction of the fraction format $ $
$\begin{array} {l} x = 5 \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } } { \color{#FF6800}{ 1 } } } \end{array}$
$\begin{array} {l} x = 5 \\ x = \color{#FF6800}{ \dfrac { \color{#FF6800}{ 3 } } { \color{#FF6800}{ 1 } } } \end{array}$
$ $ Reduce the fraction to the lowest term $ $
$\begin{array} {l} x = 5 \\ x = \color{#FF6800}{ 3 } \end{array}$