Question

If \[c > 0\]and \[4a + c < 2b\], then \[a{x^2} - bx + c = 0\]has a root in the interval
A. \[\left( {0,2} \right)\]
B. \[\left( {2,4} \right)\]
C. \[\left( { - 2,0} \right)\]
D. \[\left( {4,9} \right)\]

Answer 1

Hint: First we draw the graph of quadratic equation satisfying the above condition and also observe the value of x for which the above condition is valid. Hence, if in the stated interval the graph is cutting x-axis then it will be its root lying in that interval. And hence we can predict our answer from there.

Complete step by step answer:

As the given conditions are \[c > 0\] and \[4a + c < 2b\] and the quadratic equation is \[a{x^2} - bx + c = 0\]
Let \[f(x) = a{x^2} - bx + c = 0\],
On substituting \[x = 0\], we get,
\[f(0) = c\], and as \[c > 0\], so we have \[f\left( 0 \right) > 0\]
On substituting \[x = 2\] , we get
\[f(2) = 4a - 2b + c\], and as we have \[4a + c < 2b\]
So we get, \[f(2) < - 2b + 2b\], i.e., \[f\left( 2 \right) < 0\]
So, it is clear that \[f\left( 0 \right) > 0\]while on substituting the value of \[x = 2\]we can state that \[f\left( 2 \right) < 0\].
And hence, making the graph as

We can see that the graph change its sign in the given term and hence it’s one root lies between \[\left( {0,2} \right)\].
Hence, option (A) is correct answer.

Note: Roots are also called x-intercepts or zeros. The roots of a function are the x-intercepts. By definition, the y-coordinate of points lying on the x-axis is zero. Therefore, to find the roots of a quadratic function, we set \[f\left( x \right) = 0\]. Hence, draw the graph and examine the question stated above properly. Always remember that if for an interval, if on substituting the value of extreme ends, the nature of function changes, then root must lie between that interval, else if the nature of the function is the same then we should not conclude that no root lies between them, instead we should take smaller interval and recheck for it.