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Quadratic Polynomial Help! (1 Viewer)
- Thread starter JasonNg
- Start date
if x^2 + 2x + k = 0 has real and distinct roots with k>0, then the discriminant must be positive
.'. 4- 4k > 0
4k < 4
k < 1, but k is also >0
.'. 0< k< 1
looking at the next polynomial, the discriminant of x^2 + 2kx + 1 = 0 is 4k^2 - 4
since k is between 0 and 1, when you square k, it becomes a smaller number, but still between 0 and 1, i.e. a proper fraction
4*k^2 therefore ends up being less than 4 since multiplying by a fraction makes the number smaller
as a result 4k^2 < 4
.'. 4k^2 - 4 < 0
since the discriminant is less than 0, the polynomial x^2 + 2kx + 1 = 0 has imaginary roots</k<1
.'. 4- 4k > 0
4k < 4
k < 1, but k is also >0
.'. 0< k< 1
looking at the next polynomial, the discriminant of x^2 + 2kx + 1 = 0 is 4k^2 - 4
since k is between 0 and 1, when you square k, it becomes a smaller number, but still between 0 and 1, i.e. a proper fraction
4*k^2 therefore ends up being less than 4 since multiplying by a fraction makes the number smaller
as a result 4k^2 < 4
.'. 4k^2 - 4 < 0
since the discriminant is less than 0, the polynomial x^2 + 2kx + 1 = 0 has imaginary roots</k<1
