SOLUTION: Which of the following equation has greatest and smallest number of real solutions? (a) x^3 + x - 10 = 0 (b) x^2 + 4x - 15 = 0 (c) e^x - x = 0 (d) sec(x) - e^(-x^2) Note: Her

Algebra ->  Equations -> SOLUTION: Which of the following equation has greatest and smallest number of real solutions? (a) x^3 + x - 10 = 0 (b) x^2 + 4x - 15 = 0 (c) e^x - x = 0 (d) sec(x) - e^(-x^2) Note: Her      Log On


   

Question 1181072: Which of the following equation has greatest and smallest number of real solutions?
(a) x^3 + x - 10 = 0
(b) x^2 + 4x - 15 = 0
(c) e^x - x = 0
(d) sec(x) - e^(-x^2)
Note: Here ^ means power.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Which of the following equation has greatest and smallest number of real solutions?
(a) x%5E3+%2B+x+-+10+=+0...factor
x%5E3+%2B2x%5E2-2x%5E2%2B5x-4x+-+10+=+0
%28x%5E3+-2x%5E2%29%2B%282x%5E2-4x%29+%2B%285x-+10%29+=+0
x%5E2%28x+-2%29%2B2x%28x-2%29+%2B5%28x-+2%29+=+0
%28x+-+2%29+%28x%5E2+%2B+2x+%2B+5%29+=+0
one real solution is:x=2
find two more solutions for %28x%5E2+%2B+2x+%2B+5%29+=+0
use discriminant b%5E2-4ac, if +b%5E2-4ac%3E0 there will be real solutions
in your case a=1, b=2, c=5
+2%5E2-4%2A1%285%29%3E0
+4-20%3E0
+-16%3E0-> false, so other two solutions are complex solutions


(b) x%5E2+%2B+4x+-+15+=+0
use discriminant b%5E2-4ac, if +b%5E2-4ac%3E0 there will be real solutions
in your case a=1, b=4, c=-15
+4%5E2-4%2A1%28-15%29%3E0
+16%2B60%3E0
+76%3E0->true, so there are highlight%282%29 real solutions
(c) e%5Ex+-+x+=+0
e%5Ex+=x+....take natural log of both sides
ln%28e%5Ex%29+=ln%28x+%29
x%2Aln%28e%29+=ln%28x+%29
x%2A1+=ln%28x+%29
x=ln%28x+%29->highlight%28No%29 solution for x in R
(d) sec%28x%29+-+e%5E%28-x%5E2%29
set equal to zero
sec%28x%29+-+e%5E%28-x%5E2%29=0
sec%28x%29+=e%5E%28-x%5E2%29
only if x=0-> one real solution

answer:
the equation b) x%5E2+%2B+4x+-+15+=+0 has greatest number of real solutions
the equation c) e%5Ex+-+x+=+0 has smallest number of real solutions