Q1= show that if x is real, 2(x^2) + 6x + 9 is always positive.
Q2= Solve the simultaneous equations for x,y > 0
2log y = log 2 + log x
2^y = 4^x
cheers
Hello, sparky69er!
1) Show that if is real, is always positive.
We have: .
The sum of two squares is always nonnegative.
When , the polynomial has a minimum value of 9.
2) Solve the simultaneous equations for x, y > 0
. .
Equation [1] is: .
From Equation [2]: .
Substitute into [3]: .
. . Hence: .
Substitute into [4]: .
Given a quadratic polynomial equation of the form
,
we have solutions
(This is the "quadratic formula.")
The "discriminant" is defined as . We have three different classes of solution based on the value of the discriminant:
two real, unequal roots.
one real root. (Or two equal real roots, however you wish to look at it.)
two complex conjugate roots.
In the case of D < 0 the curve never crosses the x axis. If a > 0 then this would imply that the quadratic is always positive.
-Dan