The Discriminant

Question 1

What are the meanings of:

• b^2 - 4ac = 0
• b^2 - 4ac > 0
• b^2 - 4ac < 0

Answer 1

b^2 - 4ac = 0:

• One real root

b^2 - 4ac > 0:

• Two real roots

b^2 - 4ac < 0:

• No real solutions

Question 2

Find the set of possible values of the constant k for which the equation x^2 + kx + 9 = 0 has at least one real solution

Answer 2

• b^2 - 4ac = 0
• a = 1
• b = k
• c = 9
• ( k )^2 - 4 ( 1 ) x ( 9 )
• k^2 = 36
• k = +- 6
• ( Since 36 > 0 )
• k => 6 and k <= - 6
• ( <= or => because originally b^2 - 4ac = 0 )

Question 3

Find the values of k for which the straight line

3x + 2y = k is a tangent to the circle ( x - 1 )^2 + y^2 = 13.

Answer 3

• ( Rearrange first equation to find y )
• 2y = k - 3x
• y = k / 2 - 3 / 2x
• ( Substitute y into the other equation )
• ( x - 1 )^2 + ( k / 2 - 3 / 2x )^2 = 13
• ( x^2 - 2x + 1 ) + ( k^2 / 4 - 3k / 2x + 9 / 4x^2 ) = 13
• 13 / 4x^2 + ( - 2 - 3 / 2k )x - 12 = 0
• a = 13 / 4
• b = - 2 - 3 / 2k
• c = - 12 + k^2 / 4
• b^2 - 4ac = 0
• ( - 2 - 3 / 2k )^2 - 4 x ( 13 / 4 ) x ( - 12 + k^2 / 4 ) = 0
• 4 + 6k + 9 / 4k^2 + 156 - 13k^2 / 4 = 0
• • k^2 + 6k + 160 = 0
• k^2 - 6k - 160 = 0
• k = 16 and k = - 10