Quadratic Equation and discriminant help?

Question

Given a quadratic equation px^2 + 2mx - p + 2m =0

i)Find the discriminant of this equation.
ii)Prove that the roots are rational

Answer 1

discriminant is the part of the quadratic formula under the square root

b² - 4ac

For your equation a = p , b = 2m , c = (2m-p)

b²- 4ac = (2m)² - 4(p)(2m-p) = 4m² - 8pm + 4p²
which is equal to (2m - 2p)²

the answers will be the square root of (2m-2p)²
which squares and square roots cancel
which is simply I 2m-2p I ===> which is rational if m and p are rational

So the formula would be x = ( -b +/- sqrt(b²-4ac) ) / (2a)
Since we get a perfect square under the squareroot, we will have the ratio of two integers, which is rational solution

=]

..

Answer 2

px^2 + 2mx - p + 2m =0
px² + 2mx + (2m - p) = 0 so the discriminant is
4m² - 4p(2m - p) = 4m² - 8pm + 4p² or
4(m² + p²) - 8pm

If 4(m² + p²) > 8pm then the roots are real and unique (but -not- necessarily rational unless
4(m² + p²) - 8pm is a rational square)
If 4(m² + p²) = 8pm then there is one repeated root (at -m/p)
If 4(m² + p²) < 8pm then the roots are a complex conjugate pair.

HTH

Doug

Answer 3

the discriminant is b^2 -4ac

in the equation: px^2 + 2mx - p + 2m =0
a = p b = 2m c = -p + 2m

(2m)^2 -[ 4(p)(-p+2m) ]
= 4m^2 -(-4p^2 + 8mp)
= 4m^2 + 4p^2 - 8mp

Answer 4

b^2 - 4ac
= (2m)^2 - 4*p*(-p+2m)
= 4m^2 + 4p^2 - 8mp
= 4(m - p)^2
(m - p)^2 ≥ 0 (anything squared is ≥ 0)
The roots are rational because the discriminant is not negative.

Answer 5

When the discriminant is 0, there is a double root that lies a single point on the x-axis and is real.