Question

If p and q are roots of ax² + bx + c = 0 then find the roots of the quadratic equation ax² - bx(x-1) + c(x-1)² = 0

Answer 1

Dear Student,

p + q = $\frac{-\mathrm{b}}{\mathrm{a}}$ and pq = $\frac{\mathrm{c}}{\mathrm{a}}$
Now,
ax² - bx(x-1) + c(x-1)² = 0
ax^​2 - bx² + bx + cx^​2 - 2cx + c = 0

ax^​2 + bx + c - bx² + cx^​2 - 2cx = 0

(a + b - c)x² + (b - 2c)x + c = 0

Let m and n be the roots of above equation,
m + n = $\frac{-\left(\mathrm{b}-2\mathrm{c}\right)}{\mathrm{a}+\mathrm{b}-\mathrm{c}}$ and mn = $\frac{\mathrm{c}}{\mathrm{a}+\mathrm{b}-\mathrm{c}}$
m + n = $\frac{2\mathrm{pq}+\mathrm{p}+\mathrm{q}}{1+\mathrm{p}+\mathrm{q}+\mathrm{pq}}$
mn = $\frac{\mathrm{pq}}{1+\mathrm{p}+\mathrm{q}+\mathrm{pq}}$
Solving above two equations we can get m and n
Regards