Write each of the following in the form [ k+a(x+h)^2 a_(1)-6 x^2+5 x+7 b_(1)-5 x^2-2 x+3 ]

To write each quadratic expression in the form \( k + a(x + h)^2 \), we need to complete the square for each expression.<br /><br />### For \( a_1 - 6x^2 + 5x + 7 \):<br /><br />1. Factor out the coefficient of \( x^2 \) from the terms involving \( x \):<br /> \[<br /> a_1 - 6x^2 + 5x + 7 = a_1 + 7 - 6(x^2 - \frac{5}{6}x)<br /> \]<br /><br />2. Complete the square inside the parentheses:<br /> \[<br /> x^2 - \frac{5}{6}x = \left( x - \frac{5}{12} \right)^2 - \left( \frac{5}{12} \right)^2<br /> \]<br /> \[<br /> x^2 - \frac{5}{6}x = \left( x - \frac{5}{12} \right)^2 - \frac{25}{144}<br /> \]<br /><br />3. Substitute back and simplify:<br /> \[<br /> a_1 + 7 - 6 \left( \left( x - \frac{5}{12} \right)^2 - \frac{25}{144} \right)<br /> \]<br /> \[<br /> a_1 + 7 - 6 \left( x - \frac{5}{12} \right)^2 + \frac{150}{24}<br /> \]<br /> \[<br /> a_1 + 7 - 6 \left( x - \frac{5}{12} \right)^2 + \frac{25}{4}<br /> \]<br /> \[<br /> a_1 + \frac{53}{4} - 6 \left( x - \frac{5}{12} \right)^2<br /> \]<br /><br />So, the expression is:<br />\[<br />a_1 + \frac{53}{4} - 6 \left( x - \frac{5}{12} \right)^2<br />\]<br /><br />### For \( b_1 - 5x^2 - 2x + 3 \):<br /><br />1. Factor out the coefficient of \( x^2 \) from the terms involving \( x \):<br /> \[<br /> b_1 - 5x^2 - 2x + 3 = b_1 + 3 - 5(x^2 + \frac{2}{5}x)<br /> \]<br /><br />2. Complete the square inside the parentheses:<br /> \[<br /> x^2 + \frac{2}{5}x = \left( x + \frac{1}{5} \right)^2 - \left( \frac{1}{5} \right)^2<br /> \]<br /> \[<br /> x^2 + \frac{2}{5}x = \left( x + \frac{1}{5} \right)^2 - \frac{1}{25}<br /> \]<br /><br />3. Substitute back and simplify:<br /> \[<br /> b_1 + 3 - 5 \left( \left( x + \frac{1}{5} \right)^2 - \frac{1}{25} \right)<br /> \]<br /> \[<br /> b_1 + 3 - 5 \left( x + \frac{1}{5} \right)^2 + \frac{5}{25}<br /> \]<br /> \[<br /> b_1 + 3 - 5 \left( x + \frac{1}{5} \right)^2 + \frac{1}{5}<br /> \]<br /> \[<br /> b_1 + \frac{16}{5} - 5 \left( x + \frac{1}{5} \right)^2<br /> \]<br /><br />So, the expression is:<br />\[<br />b_1 + \frac{16}{5} - 5 \left( x + \frac{1}{5} \right)^2<br />\]