$x+\left( -2x+6 \right) \times \left( x-4 \right)=1$
Simplify the expression$x-2{x}^{2}+8x+6x-24=1$
Collect like terms$15x-2{x}^{2}-24=1$
Move the constant to the left-hand side and change its sign$15x-2{x}^{2}-24-1=0$
Use the commutative property to reorder the terms$-2{x}^{2}+15x-24-1=0$
Calculate the difference$-2{x}^{2}+15x-25=0$
Change the signs on both sides of the equation$2{x}^{2}-15x+25=0$
Write the quadratic equation in the appropriate form${x}^{2}-\frac{ 15 }{ 2 }x+\frac{ 25 }{ 2 }=0$
Identify the coefficients $p$ and $q$ of the quadratic equation$\begin{array} { l }p=-\frac{ 15 }{ 2 },& q=\frac{ 25 }{ 2 }\end{array}$
Substitute $p=-\frac{ 15 }{ 2 }$ and $q=\frac{ 25 }{ 2 }$ into the PQ formula $x=-\frac{ p }{ 2 }\pm\sqrt{ {\left( \frac{ p }{ 2 } \right)}^{2}-q }$$x=-\frac{ -\frac{ 15 }{ 2 } }{ 2 }\pm\sqrt{ {\left( \frac{ -\frac{ 15 }{ 2 } }{ 2 } \right)}^{2}-\frac{ 25 }{ 2 } }$
Determine the sign of the fraction$x=\frac{ \frac{ 15 }{ 2 } }{ 2 }\pm\sqrt{ {\left( \frac{ -\frac{ 15 }{ 2 } }{ 2 } \right)}^{2}-\frac{ 25 }{ 2 } }$
Use $\frac{ -a }{ b }=\frac{ a }{ -b }=-\frac{ a }{ b }$ to rewrite the fraction$x=\frac{ \frac{ 15 }{ 2 } }{ 2 }\pm\sqrt{ {\left( -\frac{ \frac{ 15 }{ 2 } }{ 2 } \right)}^{2}-\frac{ 25 }{ 2 } }$
Simplify the complex fraction$x=\frac{ 15 }{ 4 }\pm\sqrt{ {\left( -\frac{ \frac{ 15 }{ 2 } }{ 2 } \right)}^{2}-\frac{ 25 }{ 2 } }$
Simplify the complex fraction$x=\frac{ 15 }{ 4 }\pm\sqrt{ {\left( -\frac{ 15 }{ 4 } \right)}^{2}-\frac{ 25 }{ 2 } }$
To raise a fraction to a power, raise the numerator and denominator to that power$x=\frac{ 15 }{ 4 }\pm\sqrt{ \frac{ 225 }{ 16 }-\frac{ 25 }{ 2 } }$
Subtract the fractions$x=\frac{ 15 }{ 4 }\pm\sqrt{ \frac{ 25 }{ 16 } }$
Evaluate the square root$x=\frac{ 15 }{ 4 }\pm\frac{ 5 }{ 4 }$
Write the solutions, one with a $+$ sign and one with a $-$ sign$\begin{array} { l }x=\frac{ 15 }{ 4 }+\frac{ 5 }{ 4 },\\x=\frac{ 15 }{ 4 }-\frac{ 5 }{ 4 }\end{array}$
Add the fractions$\begin{array} { l }x=5,\\x=\frac{ 15 }{ 4 }-\frac{ 5 }{ 4 }\end{array}$
Subtract the fractions$\begin{array} { l }x=5,\\x=\frac{ 5 }{ 2 }\end{array}$
The equation has $2$ solutions$\begin{align*}&\begin{array} { l }x_1=\frac{ 5 }{ 2 },& x_2=5\end{array} \\&\begin{array} { l }x_1=2.5,& x_2=5\end{array}\end{align*}$