\begin{tabular}{|l|}\hline Example \\ \hline Express \( 2 x^{2}-12 x+3 \) in the form \( p(x-q)^{2}+r \), where \( p, q \) and \( r \) are constants to De found. \\ \hline\end{tabular}

To express \( 2x^2 - 12x + 3 \) in the form \( p(x-q)^2 + r \), we first need to complete the square. 1. Factor out the coefficient of \( x^2 \) from the first two terms: \( 2(x^2 - 6x) + 3 \). 2. Complete the square inside the parentheses. The expression \( x^2 - 6x \) can be rewritten as: \( (x - 3)^2 - 9 \) since \( 9 \) is half of \( -6 \) squared. 3. Substitute back into the equation: \( 2[(x - 3)^2 - 9] + 3 \). 4. Distribute the 2: \( 2(x - 3)^2 - 18 + 3 \). 5. Combine constants: \( 2(x - 3)^2 - 15 \). Thus, we have \( p = 2 \), \( q = 3 \), and \( r = -15 \). The final expression is: \[ 2(x - 3)^2 - 15. \]