1.Let r and s be the roots of x^2-6x+2=0 Find (r-s)^2
2.Find all solutions to the equation x^2+29=10x.
3.For what values of j does the equation (2x+7)(x-5)=-43+jx have exactly one real solution?
4.What is the largest positive integer value of m such that the equation: 3x^2-mx+7=0
has no real solutions?
5.The roots of 3x^2-4x+15=0 are the same as the roots of x^2+bx+c=0 for some constants b and c Find the ordered pair (b,c)
6.The quadratic equation 2x^2+bx+18=0 has a double root. Find all possible values of b.
7. Find all values of c such that c 4
------ = -------
c-5 c-4
Here's a few
1.Let r and s be the roots of x^2-6x+2=0 Find (r-s)^2
(r - s)^2 = r^2 - 2rs + s^2 = r^2 + s^2 - 2rs (1)
The sum of the roots = 6/1 = 6
So (r+ s)^2 = r^2 + 2rs + s^2 = 36 (2)
The product of the roots = 2 = rs
So 2rs = 4
Therefore....using (2)
r^2 + 4 + s^2 = 36
r^2 + s^2 = 32
Therefore...plugging all of this into (1)....we have that
(r - s)^2 = r^2 + s^2 - 2rs = 32 - 4 = 28
2.Find all solutions to the equation x^2+29=10x.
x^2 - 10x + 29 = 0 complete the square on x
x^2 - 10x + 25 = -29+ 25
(x - 5)^2 = -4 take both roots
x - 5 = ±√-4
x - 5 = ±2i
x = 5 ± 2i
3.For what values of j does the equation (2x+7)(x-5)=-43+jx have exactly one real solution?
2x^2 - 3x - 35 = -43 + jx rearrange
2x^2 - (3 + j)x + 8 = 0
This will have one solution when the disriminant = 0 .......so....
(3 + j)^2 - 4(2)(8) = 0
(3 + j)^2 - 64 = 0
(3 + j)^2 = 64 take both roots
3 + j = ±8
3 + j = 8 or 3 + j = -8
j = 5 j = -11
4.What is the largest positive integer value of m such that the equation: 3x^2-mx+7=0
has no real solutions?
Thiw will occur when the discriminant is < 0 .....so....
m^2 - 4(3)(7) < 0
m^2 - 84 < 0
m^2 < 84
m < sqrt (84)
So
m = 9
7.
c 4
____ = ______ cross-multiply
c - 5 c - 4
c (c - 4) = 4(c - 5)
c^2 - 4c = 4c - 20
c^2 - 8c + 20 = 0 complete the square on x
c^2 - 8c + 16 = - 20 + 16
(c - 4)^2 = -4 take both roots
c - 4 = ±√-4
c - 4 = ±2i
c = 4 ±2i