If a quadratic equation is given in standard form, we can find the sum and product of the roots using coefficient of x^{2}, x and constant term.
Let us consider the standard form of a quadratic equation,
ax^{2} + bx + c = 0
(Here a, b and c are real and rational numbers)
Let α and β be the two zeros of the above quadratic equation.
Then the formula to get sum and product of the roots of a quadratic equation is,
Find the sum and the product of the roots of the following equations.
(i) x^{2}-6x+5 = 0
(ii) kx^{2}+rx+pk = 0
(iii) 3x^{2} - 5x = 0
(iv) 8x^{2}-25 = 0
Question 1 :
x^{2}-6x+5 = 0
Solution :
By comparing the given quadratic equation, with the general form of a quadratic equation
ax^{2}+bx+c = 0
we get,
a = 1, b = -6 and c = 5
Sum of roots (α+β) = -b/a ==> 6
Product of roots (αβ) = c/a ==> 5
So, sum and product of roots are 6 and 5 respectively.
Question 2 :
kx^{2}+rx+pk = 0
Solution :
By comparing the given quadratic equation, with the general form of a quadratic equation
ax^{2}+bx+c = 0
we get,
a = k, b = r and c = pk
Sum of roots (α+β) = -b/a ==> r/k
Product of roots (αβ) = c/a ==> pk/k ==> p
So, sum and product of roots are r/k and p respectively.
Question 3 :
3x^{2}-5x = 0
Solution :
By comparing the given quadratic equation, with the general form of a quadratic equation
ax^{2}+bx+c = 0
we get,
a = 3, b = -5 and c = 0
Sum of roots (α+β) = -b/a ==> 5/3
Product of roots (αβ) = c/a ==> 0/3 ==> 0
So, sum and product of roots are 5/3 and 0 respectively.
Question 4 :
8x^{2}-25 = 0
Solution :
By comparing the given quadratic equation, with the general form of a quadratic equation
ax^{2}+bx+c = 0
we get,
a = 8, b = 0 and c = -25
Sum of roots (α+β) = -b/a ==> 0/8 ==> 0
Product of roots (αβ) = c/a ==> -25/8
So, sum and product of roots are 0 and -25/8 respectively.
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