Question

Find the values of m for which the quadratic equation x^2-m(2x-8)-15 = 0 has equal roots or both roots positive

Answer 1

Both roots are equal , it means discriminant of quadratic equation is zero.
given quadratic equation , x² - m(2x - 8) - 15 = 0
x² - 2mx + 8m - 15 = 0
x² - 2mx + (8m - 15) = 0
Now, discriminant , D = (-2m)² - 4(8m - 15) = 0
4m² - 32m + 60 = 0
m² - 8m + 15 = 0
m² - 5m - 3m + 15 = 0
(m - 5)(m - 3) = 0
m = 5 and 3

Case 1 :- take m = 5
x² - 5(2x -8) - 15 = 0
x² -10x + 40 - 15 = 0
x² - 10x + 25 = 0
Both roots are positive and x = 5
Hence, m = 5 is possible value .

Case 2 :- take m = 3
x² - 3(2x - 8) - 15 = 0
x² - 6x + 24 - 15 = 0
x² - 6x + 9 = 0
Both roots are positive and x = 3
so, m = 3 is also possible value.

Hence, m = 3 and 5

Answer 2

Hey! ! !

Solution :-

☆ i) In the first for discriminant condition

here a = 1; b = -2m and c = 8m - 15 

So

● its D is 4m² - 4(8m -15) > 0 

==> m² - 8m + 15 > 0 

==> (m - 3)(m - 5) > 0 

==> m in (-∞, 3) U (5, ∞) ---------- (1) 

● ii) Using 2nd condition, -b/a = 2m > 0; ==> m > 0 

and c/a = 8m - 15 > 0; ==> m > 15/8 

Intersection of these two, m > 15/8 ------ (2) 

Thus from (1) & (2) as above

m in (15/8, 3) U (5, ∞) 

NOTE: For m = 3 & 5, the equation has only

one root :- which is positive. :- =>

For m in (3, 5) there is no real root.

☆ ☆ ☆ Hop its helpful ☆ ☆ ☆

☆ Regards :- ♡♡《 Nitish kr singh 》♡♡