if a and b are the roots of the equation x^2 + 7x+12=0, then the equation whose roots are (a+b) and(ab)
Answers
If a, b are the roots of the equation x² + 7x + 12 = 0, Then the equation whose roots are (a + b) & ab is x² - 5x - 84 = 0
Given equation, x² + 7x + 12 = 0
Sum of roots = - 7/1 = - 7
Product of roots = 12/1 = 12
So, a + b = - 7
ab = 12
Now, The new equation is
( x + 7) ( x - 12) = 0
x² + 7x - 12x - 84 = 0
x² - 5x - 84 = 0
Verification :
Sum of roots from the equation is
-(-5/1)= 5
Sum of roots = 12 - 7 = 5
Product of roots from the equation is
-84/1 = - 84
Product of roots = 12 ( - 7) = 84
Required equation is x² - 5x - 84 = 0
Answer :-
Quadratic equation is x² - 5x - 84 = 0.
Explanation :-
x² + 7x + 12 = 0
Finding the value of a + b and ab
Given a and b are roots of the equation
Comparing ax² + bx + c = 0 with x² + 7x + 12
• a = 1
• b = 7
• c = 12
Sum of roots = a + b = - b/a
⇒a + b = - 7/1 = - 7
Product of roots = ab = c/a
⇒ab = 12/1 = 12
Finding the equation when roots of the equation are (a + b) and ab
Here
• α = (a + b) = - 7
• β = ab = 12
Sum of roots = α + β = - 7 + 12 = 5
Product of roots = αβ = - 7(12) = - 84
Quadratic equation ax² + bx + c = 0
⇒x² - x(α + β) + αβ = 0
⇒x² - x(5) + (-84) = 0
⇒x² - 5x - 84 = 0
∴ the quadratic equation is x² - 5x - 84 = 0.