To find the values of [tex]\( a \)[/tex] and [tex]\( b \)[/tex] for the quadratic equation [tex]\( ax^2 - 7x + b = 0 \)[/tex] given the roots [tex]\( x = \frac{2}{3} \)[/tex] and [tex]\( x = -3 \)[/tex], we can make use of Vieta's formulas.
### Step-by-Step Solution
#### (i) Finding the value of [tex]\( a \)[/tex]:
1. Vieta's formulas tell us that the sum of the roots of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is [tex]\( -\frac{b}{a} \)[/tex].
2. For our quadratic equation [tex]\( ax^2 - 7x + b = 0 \)[/tex], the sum of the roots (denoted [tex]\( x_1 + x_2 \)[/tex]) equates to [tex]\( -\frac{-7}{a} = \frac{7}{a} \)[/tex].
3. Given the roots [tex]\( x_1 = \frac{2}{3} \)[/tex] and [tex]\( x_2 = -3 \)[/tex], calculate their sum:
[tex]\[
x_1 + x_2 = \frac{2}{3} + (-3) = \frac{2}{3} - 3 = \frac{2}{3} - \frac{9}{3} = \frac{2 - 9}{3} = -\frac{7}{3}
\][/tex]
4. Set the equation for the sum of the roots equal to [tex]\(\frac{7}{a}\)[/tex]:
[tex]\[
\frac{7}{a} = -\frac{7}{3}
\][/tex]
5. Solve for [tex]\( a \)[/tex] by cross-multiplying and simplifying:
[tex]\[
7 \times 3 = -7 \times a \implies 21 = -7a \implies a = -3
\][/tex]
Hence, the value of [tex]\( a \)[/tex] is [tex]\(-3\)[/tex].
#### (ii) Finding the value of [tex]\( b \)[/tex]:
1. Vieta's formulas also tell us that the product of the roots [tex]\( x_1 \cdot x_2 \)[/tex] for [tex]\( ax^2 + bx + c = 0 \)[/tex] is [tex]\( \frac{c}{a} \)[/tex]. For our equation, this is [tex]\(\frac{b}{a}\)[/tex].
2. Substitute our known values to find the product [tex]\(\left(\frac{2}{3} \cdot -3\right)\)[/tex]:
[tex]\[
x_1 \cdot x_2 = \frac{2}{3} \cdot -3 = -2
\][/tex]
3. Using the fact that [tex]\( x_1 \cdot x_2 = \frac{b}{a} \)[/tex]:
[tex]\[
-2 = \frac{b}{-3}
\][/tex]
4. Solve for [tex]\( b \)[/tex] by multiplying both sides by [tex]\(-3\)[/tex]:
[tex]\[
b = -2 \times -3 = 6
\][/tex]
Hence, the value of [tex]\( b \)[/tex] is [tex]\( 6 \)[/tex].
### Summary:
- The value of [tex]\( a \)[/tex] is [tex]\(-3\)[/tex].
- The value of [tex]\( b \)[/tex] is [tex]\( 6 \)[/tex].
These are the values required for the quadratic equation [tex]\( ax^2 - 7x + b = 0 \)[/tex] given the roots [tex]\( x = \frac{2}{3} \)[/tex] and [tex]\( x = -3 \)[/tex].
