To find the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] given the roots of the quadratic equation [tex]\(ax^2 - 7x + b = 0\)[/tex], we use the properties of quadratic equations and their roots.
### (i) Finding the value of [tex]\(a\)[/tex]
Given the roots [tex]\(x = \frac{2}{3}\)[/tex] and [tex]\(x = -3\)[/tex], we can use Vieta's formulas which relate the roots of a quadratic equation to its coefficients. According to Vieta's formulas, for the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex]:
1. The sum of the roots [tex]\(\alpha + \beta = -\frac{b}{a}\)[/tex]
2. The product of the roots [tex]\(\alpha \beta = \frac{c}{a}\)[/tex]
For our quadratic equation [tex]\(ax^2 - 7x + b = 0\)[/tex]:
1. Sum of the roots: [tex]\(\frac{2}{3} + (-3) = \frac{2}{3} - \frac{9}{3} = -\frac{7}{3}\)[/tex]
2. Product of the roots: [tex]\(\left(\frac{2}{3}\right) \cdot (-3) = -2\)[/tex]
By Vieta's formulas, the sum of the roots [tex]\(\frac{2}{3} + (-3)\)[/tex] should equal [tex]\(-\frac{-7}{a}\)[/tex]:
[tex]\[
-\frac{-7}{a} = -\frac{7}{3}
\][/tex]
Simplifying:
[tex]\[
\frac{7}{a} = \frac{7}{3}
\][/tex]
By cross-multiplying and solving for [tex]\(a\)[/tex]:
[tex]\[
7 \cdot 3 = 7 \cdot a \implies 21 = 7a \implies a = 3
\][/tex]
So, the value of [tex]\(a\)[/tex] is [tex]\(\boxed{3}\)[/tex].
### (ii) Finding the value of [tex]\(b\)[/tex]
Now we use the product of the roots to find [tex]\(b\)[/tex]. According to Vieta's formulas, the product of the roots [tex]\(\frac{2}{3} \cdot (-3)\)[/tex] should equal [tex]\(\frac{b}{a}\)[/tex]:
[tex]\[
\left(\frac{2}{3}\right) \cdot (-3) = \frac{b}{3}
\][/tex]
From the previous calculation, [tex]\(\left(\frac{2}{3} \cdot (-3)\right) = -2\)[/tex]:
[tex]\[
-2 = \frac{b}{3}
\][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[
b = -2 \cdot 3 = -6
\][/tex]
So, the value of [tex]\(b\)[/tex] is [tex]\(\boxed{-6}\)[/tex].
### Final Answer:
(i) The value of [tex]\(a\)[/tex] is [tex]\(\boxed{3}\)[/tex].
(ii) The value of [tex]\(b\)[/tex] is [tex]\(\boxed{-6}\)[/tex].
