Question

If c > 0 and 4a + c < 2b, then $a{x}^2-bx+c=0$ has a root in which one of the following intervals?

• A. $(0,2)$
• B. $(2,3)$
• C. $(3,4)$
• D. $(-2,0)$

Answer 1

The correct option is A $(0,2)$

Let $f\left(x\right)=a{x}^2-bx+c$

Putting $x=0$ we get $f\left(x\right)=c$

and we know that $c>0$

therefore $f\left(x\right)$ is $positive$ at $x=0$

Now Putting $x=2$ we get $f\left(x\right)=4a-2b+c$

which can also be written as $f\left(x\right)=(4a+c)-\left(2b\right)$

and we know that $4a+c<2b$

therefore $f\left(x\right)$ is $negative$ at $x=2$

This implies that between $x=0$ and $x=2$ there exist some value of $x$ at which $f\left(x\right)=0$ which is the root of the Equation.

So a root must lie between $x=0$ and $x=2$.

In Other Options $f\left(x\right)$ is not changing its sign in between the given values.

Therefore Answer is $\left(A\right)$