what is the answer in my question?
what is the answer in my question?
Should the question say "the equations whose roots are the reciprocals of the roots of 2x^2-3x-5 = 0"?
If so, we first want to find the roots of 2x^2-3x-5=0. We factor and find two real roots (2x-5)(x+1).
Set each of these equal to zero to solve for the roots:
2x-5 = 0
2x = 5
x = 5/2
x+1 = 0
x = -1
now we want to find the reciprocals of the roots x = 5/2 and x = -1
Recall that the reciprocal of x = 1/x
Apply this rule to the roots we get
1/x = 1/(5/2) = 2/5 and
1/x = 1/-1 = -1
Now lets find the equation whose roots are equal to 2/5 and -1
We use the equation x^2-Sx+P where S = sum of the roots and P = product of the roots
S = 2/5 - 1 = 2/5 - 5/5 = -3/5
P = -2/5
So, x^2 + 3/5x - 2/5 = 5x^2 + 3x - 2
Check:
5x^2 + 3x - 2 = 0
(x+1)(5x-2) = 0
(x+1) = 0
x = -1
(5x-2) = 0
5x = 2
x = 2/5
QED.
2x^{2}-3x-5 is the equation of a parabola. Where the parabola cuts or touches the x axis are the 'roots'.
we can find these roots if they exist by 3 methods.
1. factoring the equation
2. completing the square
3. using the quadratic formula
by factoring we arrive at (2x-5)(x+1)=0 (multiply the factors together and you will get the original equation)
to get the roots we set each factor = 0
sp 2x-5=0 and x+1=0
which gives us 2x=5 and finally x = 5/2 also x = -1
the reciprocals of the roots are 2/5 and -1
now, if the roots are 2/5 and -1 then (x-2/5) and (x+1) are the factors.
if we multiply them together we get x^{2}+3/5x-2/5)=0 as the needed equation.