Find the Values of a and b from Given Roots
Question:
If \[ x=\frac{2}{3} \quad \text{and} \quad x=-3 \] are the roots of the equation \[ ax^2+7x+b=0, \] find the values of \(a\) and \(b\).
Solution
Given roots:
\[ \alpha=\frac{2}{3}, \qquad \beta=-3 \]
For the quadratic equation \[ ax^2+7x+b=0, \]
\[ \alpha+\beta=-\frac{7}{a} \] and \[ \alpha\beta=\frac{b}{a} \]
Using Sum of Roots
\[ \frac{2}{3}+(-3) = \frac{2-9}{3} = -\frac{7}{3} \]
\[ -\frac{7}{3} = -\frac{7}{a} \]
\[ a=3 \]
Using Product of Roots
\[ \frac{2}{3}\times(-3) = -2 \]
\[ -2 = \frac{b}{a} \]
Since \(a=3\),
\[ \frac{b}{3}=-2 \]
\[ b=-6 \]
Answer
Therefore,
\[ \boxed{a=3,\quad b=-6} \]
Verification:
\[ 3x^2+7x-6=0 \]
\[ (3x-2)(x+3)=0 \]
Roots: \[ x=\frac{2}{3},\,-3 \] which confirms the result.