Answer to Question #125170 in Calculus for EMMANUEL

Question

Two functions f and g are defined on the set R of real numbers by
f(x) = ax^2 + bx − 2 and g(x) = 3x + 2. If f(1) = 7 and f(2) = 10. Find:
(a) the values of a and b;
(b) the value(s) of x if f(x) = g(x) − 1.

Answer 1

(a) $f(1)=7$ and $f(2)=10$ give $a+b-2=7$ and $4a+2b-2=10$. By the first equation, $b=9-a$, so substituting this to the second one gives

$4a+2(9−a)=12⇒a+9=6⇒a=−3,b=12$.

(b) We are to solve the quadratic equation

$-3x^2+12x-2=3x+1\Leftrightarrow 0=3x^2-9x+3\Rightarrow x=\frac{3\pm\sqrt5}{2}$