# Evaluate: x-2(x-3) * (x-4)=1

## Expression: $x-2\left( x-3 \right) \times \left( x-4 \right)=1$

Distribute $-2$ through the parentheses

$x+\left( -2x+6 \right) \times \left( x-4 \right)=1$

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

$\begin{array} { l }x=\frac{ 15 }{ 4 }+\frac{ 5 }{ 4 },\\x=\frac{ 15 }{ 4 }-\frac{ 5 }{ 4 }\end{array}$

$\begin{array} { l }x=5,\\x=\frac{ 15 }{ 4 }-\frac{ 5 }{ 4 }\end{array}$
$\begin{array} { l }x=5,\\x=\frac{ 5 }{ 2 }\end{array}$
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*}