SOLUTION: Given one of the roots of the equation 2x power of 2 + 1 = h-5x is four times to other root. Find the value oh h

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Given one of the roots of the equation 2x power of 2 + 1 = h-5x is four times to other root. Find the value oh h      Log On


   

Question 1107265: Given one of the roots of the equation 2x power of 2 + 1 = h-5x is four times to other root. Find the value oh h
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
(THIS WORK CONTAINS ARITHMETIC MISTAKE)

2x%5E2%2B1=h-5x
2x%5E2%2B5x%2B1-h=0

Roots are p and v.
(BETTER CHOICE MIGHT BE ROOTS ARE p AND 4p )

p%2Bv=-5%2F2, and pv=%281-h%29%2F2
(using formulas for sum and product of roots of quadratic equation).

Let v be the larger sized root.
v=4p

Question is, solve for h:
system%28p%2Bv=-5%2F2%2C2pv=1-h%2Cv=4p%29
.
.
.
cross%28h=4%29

Answer by ikleyn(52778) About Me  (Show Source):
You can put this solution on YOUR website!
.
Given one of the roots of the equation 2x power of 2 + 1 = h-5x is four times to other root. Find the value oh h
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

The given equation

2x%5E2+%2B+1 = h-5x

is the same as

2x%5E2+%2B+5x+%2B+%281-h%29 = 0,

which is equivalent to

x%5E2+%2B+%285%2F2%29%2Ax+%2B+%281-h%29%2F2 = 0.     (1)


Let one root be r.

Then the other root is 4r, according to the condition.


The sum of the roots is equal to the coefficient at "x" in the equation (1) taken with the opposite sign (the Vieta's theorem). Hence

r + 4r = -5%2F2,   or   5r = -5%2F2,   which implies  r = -1%2F2.


Now substitute -1%2F2 into equation (1) and take into account that -1%2F2 is the root. You will get

%28-1%2F2%29%5E2+%2B+%285%2F2%29%2A%28-1%2F2%29+%2B+%281-h%29%2F2 = 0.


It is your equation to find h.   Simplify and solve for h:

1%2F4 - 5%2F4 = %28h-1%29%2F2,

-1 = %28h-1%29%2F2,

h-1 = -2  ====>  h = -2 + 1 = -1.


Answer.  h = -1.

Solved.

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Ignore the solution by @josgaritmetic, since it leads to nowhere.