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If a,b,c € R, a not equals to 0 and the quadratic equation,ax 2 + bx + C = 0 has no real root, then show that a(a+b+c)>0

If a,b,c € R, a not equals to 0 and the quadratic equation,ax2 + bx + C = 0 has no real root, then show that a(a+b+c)>0

Grade:12

1 Answers

Aditya Gupta
2081 Points
5 years ago
this is an easy ques based on graph of a quadratic function.
assume that a is positive.
then the graph of y= f(x)= ax2 + bx + c would be above the x axis since it has no real root.
so f(1) is greater than zero.
but f(1)= a+b+c
so a+b+c is greater than zero.
so a(a+b+c) is greater than zero.
now assume that a is less than zero. 
then the graph of y= f(x)= ax2 + bx + c would be below the x axis since it has no real root.
so f(1) is less than zero.
so a+b+c is less than zero.
so a(a+b+c) is greater than zero.

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