Sure! Let's solve the problem step by step based on the given information that [tex]\( x = \frac{2}{3} \)[/tex] and [tex]\( x = -3 \)[/tex] are the roots of the quadratic equation [tex]\( a x^2 - 7 x + b = 0 \)[/tex].
### Step-by-Step Solution:
#### (i) Finding the value of [tex]\( a \)[/tex]:
To find the value of [tex]\( a \)[/tex], we can use Vieta's formulas which relate the coefficients of a polynomial to sums and products of its roots:
1. Sum of the Roots: For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]:
- The sum of the roots [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex] is given by [tex]\( \frac{-b}{a} \)[/tex].
Given roots:
[tex]\[ r_1 = \frac{2}{3} \][/tex]
[tex]\[ r_2 = -3 \][/tex]
The sum of the roots:
[tex]\[ r_1 + r_2 = \frac{2}{3} + (-3) = \frac{2}{3} - 3 = \frac{2}{3} - \frac{9}{3} = \frac{2 - 9}{3} = \frac{-7}{3} \][/tex]
So,
[tex]\[ \frac{-b}{a} = \frac{-(-7)}{a} = \frac{7}{a} \][/tex]
Thus,
[tex]\[ \frac{-7}{3} = \frac{7}{a} \][/tex]
By cross-multiplying to solve for [tex]\( a \)[/tex]:
[tex]\[ a \times -7 = 7 \times 3 \][/tex]
[tex]\[ -7a = 21 \][/tex]
[tex]\[ a = \frac{21}{-7} \][/tex]
[tex]\[ a = -3 \][/tex]
Therefore, the value of [tex]\( a \)[/tex] is:
[tex]\[ a = -3 \][/tex]
#### (ii) Finding the value of [tex]\( b \)[/tex]:
To find the value of [tex]\( b \)[/tex], we can use the product of the roots:
2. Product of the Roots: For a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex]:
- The product of the roots [tex]\( r_1 \)[/tex] and [tex]\( r_2 \)[/tex] is given by [tex]\( \frac{c}{a} \)[/tex].
The product of the roots:
[tex]\[ r_1 \times r_2 = \frac{2}{3} \times (-3) = \frac{2 \times (-3)}{3} = \frac{-6}{3} = -2 \][/tex]
So,
[tex]\[ \frac{c}{a} = \frac{b}{a} \][/tex]
Thus,
[tex]\[ -2 = \frac{b}{a} \][/tex]
Given that [tex]\( a = -3 \)[/tex],
[tex]\[ -2 = \frac{b}{-3} \][/tex]
By cross-multiplying to solve for [tex]\( b \)[/tex]:
[tex]\[ -2 \times -3 = b \][/tex]
[tex]\[ 6 = b \][/tex]
Therefore, the value of [tex]\( b \)[/tex] is:
[tex]\[ b = 6 \][/tex]
### Summary:
(i) The value of [tex]\( a \)[/tex] is [tex]\( \boxed{-3} \)[/tex].
(ii) The value of [tex]\( b \)[/tex] is [tex]\( \boxed{6} \)[/tex].
