if α and β are zeroes of quadratic polynomial f(x) x²-7x+p,such that α²+β²=29,find the value of p
Answers
Answer:
Step-by-step explanation:
This problem is a simple lesson in Vieta's formulas which relate sums of products of roots to the coefficients in a polynomial. Note these formulas apply to any degree polynomial, but here we only need pay attention to quadratics.
Start by writing out the general factored form of the quadratic with roots α and β. Then proceed with expanding the quadratic.
c(x−α)(x−β)=cx2−c(α+β)x+cαβ
We can equate this to the original quadratic.
cx2−c(α+β)x+cαβ=x2+7x+10
Compare the x2 coefficients pairwise so c=1. Similarly, -7=α+β and 10=αβ. Therefore we can answer the question as 10+−7=3 without explicitly knowing the roots.
And there you have it, Vieta’s formulas for quadratics. You can also generalize to higher degree polynomials. The pattern becomes easy to notice if you expand a cubic function
Answer:See Image.
Step-by-step explanation:
