Consider the quadratic function in $x$ $$y = a x ^ { 2 } + b x + c .$$ The function (1) takes its maximum value 16 at $x = 1$, its graph intersects the $x$-axis at two points, and the length of the segment connecting those two points is 8. We are to find the values of $a$, $b$ and $c$. From the conditions, (1) can be represented as $$y = a ( x - \mathbf { A } ) ^ { 2 } + \mathbf { B } \mathbf { C }$$ and the coordinates of the two points at which the graph of (1) and the $x$-axis intersect are $$( - \mathbf { D } , 0 ) , \quad ( \mathbf { E } , 0 ) .$$ Thus we obtain $a = \mathbf { F G }$. Hence we have $$b = \mathbf { H } , \quad c = \mathbf { I J } .$$
Consider the quadratic function in $x$
$$y = a x ^ { 2 } + b x + c .$$
The function (1) takes its maximum value 16 at $x = 1$, its graph intersects the $x$-axis at two points, and the length of the segment connecting those two points is 8. We are to find the values of $a$, $b$ and $c$.
From the conditions, (1) can be represented as
$$y = a ( x - \mathbf { A } ) ^ { 2 } + \mathbf { B } \mathbf { C }$$
and the coordinates of the two points at which the graph of (1) and the $x$-axis intersect are
$$( - \mathbf { D } , 0 ) , \quad ( \mathbf { E } , 0 ) .$$
Thus we obtain $a = \mathbf { F G }$. Hence we have
$$b = \mathbf { H } , \quad c = \mathbf { I J } .$$