Question 259039
Lets take one at a time:
a) What is the value of k if x=3 is a root of 2x^2 - kx - 3=0 ?
x = 3 is a root means that (x-3)=0.
Now, (x-3)(2x+1) = 2x^2 -5x -3, so k = 5.
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b) Determine the value(s) of k for which the equation kx^2 -4x + k = 0 will have two equal and real roots.
If a quadratic has two equal roots the discriminant, or D = b^2-4ac = 0.
Applying this to our function, we get
{{{d = (-4)^2 - 4*k*k}}}
or
16-4k^2 = 0
16 = 4k^2
4 = k^2
K = +-2
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c) What is the nature of the roots of 2x^2 - x - 3 = -5? 
When they say nature of the roots, we are going after the quadratic formula which is
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
based on our polynomial, we get
{{{x = (1 +- sqrt( (-1)^2-4*2*(-3) ))/(2*2) }}}
which is
{{{x = (1 +- sqrt( 25))/(4) }}}
or
{{{x = (1 +-5)/4}}}
we have two real rational roots.