If both the roots - - of the quadratic equation ax2-bx-c-0- a-Considering fx-ax2-bx-c- select the correct statements-

Given the quadratic polynomial: f(x)=ax^2+bx+c, a lt;0 Let α, β be it's roots of f(x)=0. Now, k is a real number such that k gt;α gt;β So, we can draw the ...

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Byju's Answer

Standard XII

Mathematics

Location of Roots when Compared with a constant 'k'

If both the r...

Question

If both the roots $α, β$ of the quadratic equation $a x^{2} + b x + c = 0, a < 0$ are less than a real number $k$ such that $k > α > β$
Considering $f (x) = a x^{2} + b x + c$ , select the correct statement(s).

f (k) < 0

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k < - \frac{b}{2 a}

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D > 0

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f (k) > 0

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Solution

The correct option is C $D > 0$
Given the quadratic polynomial: $f (x) = a x^{2} + b x + c, a < 0$
Let $α, β$ be it's roots of $f (x) = 0$ .
Now, $k$ is a real number such that $k > α > β$
So, we can draw the graph as:

Clearly, for this to happen, the following conditions needs to be satisfied:
$(a) D > 0$ because $f (x) = 0$ has $2$ distinct roots.
$(b) f (k) < 0$ .
$(c)$ Abscissa of Vertex $= \frac{- b}{2 a} < k$

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If both the roots α,β of the quadratic equation ax2+bx+c=0,a<0 are less than a real number k such that k>α>β Considering f(x)=ax2+bx+c, select the correct statement(s).

If both the roots $α, β$ of the quadratic equation $a x^{2} + b x + c = 0, a < 0$ are less than a real number $k$ such that $k > α > β$
Considering $f (x) = a x^{2} + b x + c$ , select the correct statement(s).