Question

For what values of c does the quadratic equation
`x^2-2x+c=0` have:

a. no real roots
b. two roots of the same sign
c. one root equal to zero and one negative root
d. two roots of opposite signs

Answer 1

Explanation:

a: in the quadratic formula, no real roots would mean b^2 - 4ac<0 because of the discriminant. b^2 is 4, so all 4c has to satisfy is that it’s greater than 4. Solving, c>1

b: casework:

Both are negative: This is impossible because square roots are always positive, so at least one would always be positive.

Both are positive: sqrt(4-4c)<2. 0<c<=1.

c. I’m not entirely sure of what you mean, but when c=0, 0 is a root of the quadratic equation, but the other root is positive 2, so no value?

d. sqrt(4-4c)>2. Another possibility is that the same thing is less than -2, but square roots are always positive. This remains true for c<0.