Question

If a quadratic expression cuts the $x-$ axis at $x=\alpha \&\beta$ and it's vertex is given as: $(p,q),$ then $p=$

• A. $\alpha +\beta$
• B. $-\frac{\alpha +\beta }{2}$
• C. $\frac{\alpha +\beta }{2}$

Answer 1

The correct option is C $\frac{\alpha +\beta }{2}$

Let $f\left(x\right)=a{x}^2+bx+c$
such that $\alpha ,\beta$ are the two roots of $f\left(x\right)=0$.
And $(p,q)$ is it's vertex.
Since, vertex is given by $V\equiv (\frac{-b}{2a},\frac{-D}{4a})$
Also, if $\alpha ,\beta$ are it's roots.
Then, Sum of roots $=\alpha +\beta =-\frac{b}{a}$
$\Rightarrow \frac{\alpha +\beta }{2}=-\frac{b}{2a}$
Which is equal to the $x-$ coordinate of the vertex.

Since $(p,q)$ is the coordinates of vertex for given expression.
Then, $p=\frac{\alpha +\beta }{2}$