Samacheer Kalvi Class 10 Maths Solutions Chapter 3 Algebra Exercise 3.9

Question 1. Determine the quadratic equations, whose sum and product of roots are
(i) -9, 20
(ii) \( \frac { 5 }{ 3 } \), 4
(iii) \( \frac { -3 }{ 2 } \), -1
(iv) \( -(2 - a)^2 \), \( (a + 5)^2 \)
Answer:
To find the quadratic equation when the sum (\( \alpha + \beta \)) and product (\( \alpha\beta \)) of its roots are known, we use the general formula: \( x^2 - (\alpha + \beta)x + \alpha\beta = 0 \). This formula helps construct the equation directly from its roots' properties.
(i) Given sum of roots \( (\alpha + \beta) = -9 \) and product of roots \( (\alpha\beta) = 20 \).
Substitute these values into the formula:
\( x^2 - (-9)x + 20 = 0 \)
\( \implies x^2 + 9x + 20 = 0 \)
(ii) Given sum of roots \( (\alpha + \beta) = \frac { 5 }{ 3 } \) and product of roots \( (\alpha\beta) = 4 \).
Substitute these values into the formula:
\( x^2 - (\frac { 5 }{ 3 } )x + 4 = 0 \)
To clear the fraction, multiply the entire equation by 3:
\( \implies 3x^2 - 5x + 12 = 0 \)
(iii) Given sum of roots \( (\alpha + \beta) = \frac { -3 }{ 2 } \) and product of roots \( (\alpha\beta) = -1 \).
Substitute these values into the formula:
\( x^2 - (\frac { -3 }{ 2 } )x + (-1) = 0 \)
\( \implies x^2 + \frac { 3 }{ 2 } x - 1 = 0 \)
To clear the fraction, multiply the entire equation by 2:
\( \implies 2x^2 + 3x - 2 = 0 \)
(iv) Given sum of roots \( (\alpha + \beta) = -(2 - a)^2 \) and product of roots \( (\alpha\beta) = (a + 5)^2 \).
Substitute these values into the formula:
\( x^2 - [-(2 - a)^2]x + (a + 5)^2 = 0 \)
\( \implies x^2 + (2 - a)^2 x + (a + 5)^2 = 0 \)
In simple words: We use a special formula that connects the sum and product of roots directly to the quadratic equation. We put the given sum and product values into this formula to get the final equation. Sometimes, we multiply the whole equation by a number to remove any fractions.

🎯 Exam Tip: Always remember the general form of a quadratic equation: \( x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \). Pay close attention to signs, especially when the sum of roots is negative.

Question 2. Find the sum and product of the roots for each of the following quadratic equations
(i) \( x^2 + 3x - 28 = 0 \)
(ii) \( x^2 + 3x = 0 \)
(iii) \( 3 + \frac{1}{a}=\frac{10}{a^{2}} \)
(iv) \( 3y^2 - y - 4 = 0 \)
Answer:
For a quadratic equation in the standard form \( Ax^2 + Bx + C = 0 \), the sum of the roots \( (\alpha + \beta) \) is \( -\frac{B}{A} \) and the product of the roots \( (\alpha\beta) \) is \( \frac{C}{A} \). These relationships are fundamental in understanding quadratic equations without solving for the roots themselves.
(i) Equation: \( x^2 + 3x - 28 = 0 \)
Here, comparing with \( Ax^2 + Bx + C = 0 \), we have \( A=1, B=3, C=-28 \).
Sum of the roots \( (\alpha + \beta) = -\frac{B}{A} = -\frac{3}{1} = -3 \).
Product of the roots \( (\alpha\beta) = \frac{C}{A} = \frac{-28}{1} = -28 \).
(ii) Equation: \( x^2 + 3x = 0 \)
Here, comparing with \( Ax^2 + Bx + C = 0 \), we have \( A=1, B=3, C=0 \).
Sum of the roots \( (\alpha + \beta) = -\frac{B}{A} = -\frac{3}{1} = -3 \).
Product of the roots \( (\alpha\beta) = \frac{C}{A} = \frac{0}{1} = 0 \).
(iii) Equation: \( 3 + \frac{1}{a}=\frac{10}{a^{2}} \)
First, convert this equation into the standard quadratic form \( Aa^2 + Ba + C = 0 \). Multiply the entire equation by \( a^2 \) (assuming \( a \neq 0 \)):
\( 3a^2 + a = 10 \)
\( \implies 3a^2 + a - 10 = 0 \)
Here, comparing with \( Aa^2 + Ba + C = 0 \), we have \( A=3, B=1, C=-10 \).
Sum of the roots \( (\alpha + \beta) = -\frac{B}{A} = -\frac{1}{3} \).
Product of the roots \( (\alpha\beta) = \frac{C}{A} = \frac{-10}{3} \).
(iv) Equation: \( 3y^2 - y - 4 = 0 \)
Here, comparing with \( Ay^2 + By + C = 0 \), we have \( A=3, B=-1, C=-4 \).
Sum of the roots \( (\alpha + \beta) = -\frac{B}{A} = -\frac{(-1)}{3} = \frac{1}{3} \).
Product of the roots \( (\alpha\beta) = \frac{C}{A} = \frac{-4}{3} \).
In simple words: For any quadratic equation, you can quickly find the sum and product of its solutions (roots) without actually solving the equation. Just identify the numbers in front of \( x^2 \), \( x \), and the constant term, then use the simple formulas. If the equation is not in the standard form, change it first.

🎯 Exam Tip: Always write the quadratic equation in the standard form \( Ax^2 + Bx + C = 0 \) before identifying A, B, and C. Remember that the sign of B is important for the sum of roots, and fractions should be simplified.