Let $a$ and $b$ be real numbers, where $a > 0$. Consider the two quadratic functions $$f ( x ) = 2 x ^ { 2 } - 4 x + 5 , \quad g ( x ) = x ^ { 2 } + a x + b .$$ We are to find the values of $a$ and $b$ when the function $g ( x )$ satisfies the following two conditions. (i) The minimum value of $g ( x )$ is 8 less than the minimum value of $f ( x )$. (ii) There exists only one $x$ which satisfies $f ( x ) = g ( x )$. Since the minimum value of $f ( x )$ is $\mathbf{A}$, from condition (i), we derive the equality $$b = \frac { a ^ { 2 } } { \mathbf { B } } - \mathbf { C } \text {. }$$ Hence the equation from which we can find the $x$ satisfying $f ( x ) = g ( x )$ is $$x ^ { 2 } - ( a + \mathbf { D } ) x - \frac { a ^ { 2 } } { \mathbf { E } } + \mathbf { F G } = 0 .$$ Thus, since $a > 0$, from condition (ii) we obtain $$a = \mathbf { H } , \quad b = \mathbf { I J } .$$ In this case, the $x$ satisfying $f ( x ) = g ( x )$ is $\mathbf { K }$.
Let $a$ and $b$ be real numbers, where $a > 0$. Consider the two quadratic functions
$$f ( x ) = 2 x ^ { 2 } - 4 x + 5 , \quad g ( x ) = x ^ { 2 } + a x + b .$$
We are to find the values of $a$ and $b$ when the function $g ( x )$ satisfies the following two conditions.\\
(i) The minimum value of $g ( x )$ is 8 less than the minimum value of $f ( x )$.\\
(ii) There exists only one $x$ which satisfies $f ( x ) = g ( x )$.
Since the minimum value of $f ( x )$ is $\mathbf{A}$, from condition (i), we derive the equality
$$b = \frac { a ^ { 2 } } { \mathbf { B } } - \mathbf { C } \text {. }$$
Hence the equation from which we can find the $x$ satisfying $f ( x ) = g ( x )$ is
$$x ^ { 2 } - ( a + \mathbf { D } ) x - \frac { a ^ { 2 } } { \mathbf { E } } + \mathbf { F G } = 0 .$$
Thus, since $a > 0$, from condition (ii) we obtain
$$a = \mathbf { H } , \quad b = \mathbf { I J } .$$
In this case, the $x$ satisfying $f ( x ) = g ( x )$ is $\mathbf { K }$.