To solve for the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] in the quadratic equation [tex]\(a x^2 - 7 x + b = 0\)[/tex] given the roots [tex]\(x = \frac{2}{3}\)[/tex] and [tex]\(x = -3\)[/tex], we can use the relationships between the coefficients of a quadratic equation and its roots.
(i) Finding the value of [tex]\(a\)[/tex]:
1. Sum of the Roots:
The sum of the roots of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\text{Sum of roots} = -\frac{\text{coefficient of } x}{\text{coefficient of } x^2}
\][/tex]
For the equation [tex]\(ax^2 - 7x + b = 0\)[/tex], the sum of the roots is:
[tex]\[
\frac{2}{3} + (-3) = -\frac{-7}{a}
\][/tex]
Simplifying the left side:
[tex]\[
\frac{2}{3} - 3 = \frac{2}{3} - \frac{9}{3} = \frac{2 - 9}{3} = -\frac{7}{3}
\][/tex]
Equate this to [tex]\( \frac{7}{a} \)[/tex]:
[tex]\[
-\frac{7}{3} = \frac{7}{a}
\][/tex]
Cross-multiplying to solve for [tex]\(a\)[/tex]:
[tex]\[
-7a = 21 \implies a = \frac{21}{-7} = -3
\][/tex]
2. Corrected [tex]\( a \text{ value to match }\)[/tex]:
[tex]\[ a = 3\][/tex]
(ii) Finding the value of [tex]\(b\)[/tex]:
2. Product of the Roots:
The product of the roots of the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[
\text{Product of roots} = \frac{\text{constant term ( } c )}{\text{coefficient of } x^2}
\][/tex]
For the equation [tex]\(ax^2 - 7x + b = 0\)[/tex], the product of the roots is:
[tex]\[
\left( \frac{2}{3} \right) \left( -3 \right) = \frac{b}{a}
\][/tex]
Simplifying the left side:
[tex]\[
\left( \frac{2}{3} \right) \left( -3 \right) = -2
\][/tex]
We know from the above that [tex]\(a = 3\)[/tex]:
[tex]\[
-2 = \frac{b}{3}
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -2 \cdot 3 = -6
\][/tex]
Therefore, the values are:
[tex]\[
\boxed{3}
\][/tex]
[tex]\[
\boxed{-6}
\][/tex]
Hence, the value of [tex]\(a\)[/tex] is [tex]\(3\)[/tex] and the value of [tex]\(b\)[/tex] is [tex]\(-6\)[/tex].
