Q. If the equation `ax^2-bx+5=0` doesn't have two distinct real roots then the minimum value of `a+b` is A) -5 B) 5 C) 0 D) none of these

embizze | High School Teacher | (Level 1) Educator Emeritus

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Given `ax^2-bx+5=0` :

The discriminant describes the two zeros; if the discriminant is negative then there are 0 real roots and two complex roots, if the discriminant is zero then there is exactly 1 eral root (a double root), and if the discriminant is positive then there are two distinct real roots.

For `ax^2+bx+c` the discriminant is `b^2-4ac` .

For the given equation, the discriminant of the quadratic expression is:

`(-b)^2-4(a)(5)=b^2-20a`

If there are not two distinct real roots, then the discriminant is nonpositive.

`b^2-20a<=0`

`b^2<=20a`

This inequality cannot be true for any a<0, so `a>=0` .

``If we select b>0 so that the inequality holds, the inequality will also hold for -b; the sum will be minimized if we choose the negative so `b<=0` .

Equality occurs when a=b=0 and a=|b|=20. For a>20, |b|<a so a+(-b)>0.

The minimum occurs when b=-10 and a=5.

The equation becomes `5x^2+10x+5=0 ==> 5(x+1)^2=0` which has a double root at x=-1. We see that a+b=5-10=-5.

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The answer is (A)

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