Answer :
To solve this problem, we'll start from the general properties of quadratic equations. For a quadratic equation of the form
[tex]\[ ay^2 + by + c = 0 \][/tex]
the sum and product of the roots can be derived from the coefficients:
1. The sum of the roots ([tex]\(\alpha + \beta\)[/tex]) is given by [tex]\(-\frac{b}{a}\)[/tex].
2. The product of the roots ([tex]\(\alpha \beta\)[/tex]) is given by [tex]\(\frac{c}{a}\)[/tex].
Given the equation:
[tex]\[ k y^2 - 11 y + (k - 23) = 0 \][/tex]
we can identify the coefficients as:
- [tex]\(a = k\)[/tex]
- [tex]\(b = -11\)[/tex]
- [tex]\(c = k - 23\)[/tex]
From the standard properties of quadratic equations, the sum of the roots ([tex]\(\alpha + \beta\)[/tex]) can be expressed as:
[tex]\[ \alpha + \beta = -\frac{-11}{k} = \frac{11}{k} \][/tex]
and the product of the roots ([tex]\(\alpha \beta\)[/tex]) can be expressed as:
[tex]\[ \alpha \beta = \frac{k - 23}{k} \][/tex]
According to the problem statement, the sum of the roots is [tex]\(\frac{13}{21}\)[/tex] more than the product of the roots:
[tex]\[ \alpha + \beta = \alpha \beta + \frac{13}{21} \][/tex]
Substitute the expressions for the sum and product of the roots:
[tex]\[ \frac{11}{k} = \frac{k - 23}{k} + \frac{13}{21} \][/tex]
To solve this equation, we first get rid of the fractions by multiplying through by [tex]\(k\)[/tex]:
[tex]\[ 11 = (k - 23) + \frac{13k}{21} \][/tex]
[tex]\[ 11 = k - 23 + \frac{13k}{21} \][/tex]
Combine the terms involving [tex]\(k\)[/tex] on one side:
[tex]\[ 11 = k - 23 + \frac{13k}{21} \][/tex]
To combine the terms involving [tex]\(k\)[/tex], we need to have a common denominator. Rewrite [tex]\(k\)[/tex] and 23 using 21 as the common denominator:
[tex]\[ 11 = \frac{21k - 483 + 13k}{21} \][/tex]
[tex]\[ 11 = \frac{34k - 483}{21} \][/tex]
Next, clear the fraction by multiplying both sides by 21:
[tex]\[ 11 \times 21 = 34k - 483 \][/tex]
[tex]\[ 231 = 34k - 483 \][/tex]
Add 483 to both sides to isolate [tex]\(34k\)[/tex]:
[tex]\[ 231 + 483 = 34k \][/tex]
[tex]\[ 714 = 34k \][/tex]
Divide both sides by 34:
[tex]\[ k = \frac{714}{34} \][/tex]
[tex]\[ k = 21 \][/tex]
Therefore, the value of [tex]\(k\)[/tex] is:
[tex]\[ \boxed{21} \][/tex]
[tex]\[ ay^2 + by + c = 0 \][/tex]
the sum and product of the roots can be derived from the coefficients:
1. The sum of the roots ([tex]\(\alpha + \beta\)[/tex]) is given by [tex]\(-\frac{b}{a}\)[/tex].
2. The product of the roots ([tex]\(\alpha \beta\)[/tex]) is given by [tex]\(\frac{c}{a}\)[/tex].
Given the equation:
[tex]\[ k y^2 - 11 y + (k - 23) = 0 \][/tex]
we can identify the coefficients as:
- [tex]\(a = k\)[/tex]
- [tex]\(b = -11\)[/tex]
- [tex]\(c = k - 23\)[/tex]
From the standard properties of quadratic equations, the sum of the roots ([tex]\(\alpha + \beta\)[/tex]) can be expressed as:
[tex]\[ \alpha + \beta = -\frac{-11}{k} = \frac{11}{k} \][/tex]
and the product of the roots ([tex]\(\alpha \beta\)[/tex]) can be expressed as:
[tex]\[ \alpha \beta = \frac{k - 23}{k} \][/tex]
According to the problem statement, the sum of the roots is [tex]\(\frac{13}{21}\)[/tex] more than the product of the roots:
[tex]\[ \alpha + \beta = \alpha \beta + \frac{13}{21} \][/tex]
Substitute the expressions for the sum and product of the roots:
[tex]\[ \frac{11}{k} = \frac{k - 23}{k} + \frac{13}{21} \][/tex]
To solve this equation, we first get rid of the fractions by multiplying through by [tex]\(k\)[/tex]:
[tex]\[ 11 = (k - 23) + \frac{13k}{21} \][/tex]
[tex]\[ 11 = k - 23 + \frac{13k}{21} \][/tex]
Combine the terms involving [tex]\(k\)[/tex] on one side:
[tex]\[ 11 = k - 23 + \frac{13k}{21} \][/tex]
To combine the terms involving [tex]\(k\)[/tex], we need to have a common denominator. Rewrite [tex]\(k\)[/tex] and 23 using 21 as the common denominator:
[tex]\[ 11 = \frac{21k - 483 + 13k}{21} \][/tex]
[tex]\[ 11 = \frac{34k - 483}{21} \][/tex]
Next, clear the fraction by multiplying both sides by 21:
[tex]\[ 11 \times 21 = 34k - 483 \][/tex]
[tex]\[ 231 = 34k - 483 \][/tex]
Add 483 to both sides to isolate [tex]\(34k\)[/tex]:
[tex]\[ 231 + 483 = 34k \][/tex]
[tex]\[ 714 = 34k \][/tex]
Divide both sides by 34:
[tex]\[ k = \frac{714}{34} \][/tex]
[tex]\[ k = 21 \][/tex]
Therefore, the value of [tex]\(k\)[/tex] is:
[tex]\[ \boxed{21} \][/tex]
