BÀI 5: Cho hàm số y = f(x) = ax2 + bx + c. Tìm a, b, c biết a) f(1) = 5; f( 2) = 9; f(0) = 2 b) f(1) =2; f( 2) = 4; f(-1) = 6

Đáp án:

$\begin{array}{l}
a)\left\{ \begin{array}{l}
f\left( 1 \right) = 5\\
f\left( 2 \right) = 9\\
f\left( 0 \right) = 2
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
a{.1^2} + b.1 + c = 5\\
a{.2^2} + b.2 + c = 9\\
a{.0^2} + b.0 + c = 2
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
a + b + c = 5\\
4a + 2b + c = 9\\
c = 2
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
c = 2\\
a + b = 5 - c = 3\\
4a + 2b = 9 - c = 7
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
c = 2\\
2a + 2b = 6\\
4a + 2b = 7
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
c = 2\\
2a = 1\\
b = 3 - a
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
c = 2\\
a = \dfrac{1}{2}\\
b = 3 - a = 3 - \dfrac{1}{2} = \dfrac{5}{2}
\end{array} \right.\\
Vậy\,a = \dfrac{1}{2};b = \dfrac{5}{2};c = 2\\
b)\\
\left\{ \begin{array}{l}
f\left( 1 \right) = 2\\
f\left( 2 \right) = 4\\
f\left( { - 1} \right) = 6
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
a{.1^2} + b.1 + c = 2\\
a{.2^2} + b.2 + c = 4\\
a.{\left( { - 1} \right)^2} - b + c = 6
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
a + b + c = 2\\
4a + 2b + c = 4\\
a - b + c = 6
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
2b = 2 - 6\\
a + c = 2 - b\\
4a + c = 4 - 2b
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
b =  - 2\\
a + c = 2 - \left( { - 2} \right) = 4\\
4a + c = 4 - 2.\left( { - 2} \right) = 8
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
b =  - 2\\
a + c = 4\\
4a + c = 8
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
b =  - 2\\
3a = 4\\
c = 4 - a
\end{array} \right.\\
 \Leftrightarrow \left\{ \begin{array}{l}
b =  - 2\\
a = \dfrac{4}{3}\\
c = 4 - a = \dfrac{8}{3}
\end{array} \right.\\
Vậy\,a = \dfrac{4}{3};b =  - 2;c = \dfrac{8}{3}
\end{array}$