Question

Find quadrilateral polynomial are alpha =5/2and-5/2

Answer 1

Step-by-step explanation:

Sum and product of the roots

We have seen that, in the case when a parabola crosses the x-axis, the x-coordinate of the vertex lies at the average of the intercepts. Thus, if a quadratic has two real roots α,β, then the x-coordinate of the vertex is 12(α+β). Now we also know that this quantity is equal to −b2a. Thus we can express the sum of the roots in terms of the coefficients a,b,c of the quadratic as α+β=−ba.

In the case when the quadratic does not cross the x-axis, the corresponding quadratic equation ax2+bx+c=0 has no real roots, but it will have complex roots (involving the square root of negative numbers). The formula above, and other similar formulas shown below, still work in this case.

We can find simple formulas for the sum and product of the roots simply by expanding out. Thus, if α,β are the roots of ax2+bx+c=0, then dividing by a we have

x2+bax+ca=(x−α)(x−β)=x2−(α+β)x+αβ.

Comparing the first and last expressions we conclude that

α+β=−baandαβ=ca.

From these formulas, we can also find the value of the sum of the squares of the roots of a quadratic without actually solving the quadratic.