I need a math WIZ with this problem?

Question

Unknown values: Given the equation 2x squared - kx + 16=0, for what value(s) of k would root be twice the other?

Let R represent the first root.
Let 2R represent the second root.

Sum of the roots = ??? Product of the roots = ???

Choices for answers: 2R, 3R, 2Rsquared, or 3R squared.

Answer 1

2x^2 - kx + 16 = 0

Let m = the first root and n = the second root. By the quadratic formula,

m = [-b + sqrt(b^2 - 4ac)] / (2a)
n = [-b - sqrt(b^2 - 4ac)] / (2a)
So with a = 2, b = -k, c = 16,

m = [k + sqrt(k^2 - 4(2)(16)] / 4
n = [k - sqrt(k^2 - 4(2)(16)] / 4

Simplifying some more,

m = [k + sqrt(k^2 - 128)] / 4
n = [k - sqrt(k^2 - 128)] / 4

Since one root is twice the other, it follows that
m = 2n, so

[k + sqrt(k^2 - 128)] / 4 = 2 { [k - sqrt(k^2 - 128)] / 4 }

All we have to do is solve for k. First, let's multiply by 4, to get rid of the denominator.

k + sqrt(k^2 - 128) = 2 [k - sqrt(k^2 - 128)]

Distribute the 2.

k + sqrt(k^2 - 128) = 2k - 2sqrt(k^2 - 128)

Move all the terms with k in them to the right hand side, and everything else to the left hand side.

sqrt(k^2 - 128) + 2sqrt(k^2 - 128) = k

Group like terms on the left hand side.

3sqrt(k^2 - 128) = k

Now we try and solve for k. Square both sides,

9(k^2 - 128) = k^2

9k^2 - 1152 = k^2
8k^2 = 1152
k^2 = 144, so it follows that

k = +/- 12

k = {-12, 12}

Answer 2

Solve for k in the equation below such that one root is twice the other.

2x² - kx + 16 = 0

(2x - 4)(x - 4) = 0
x = 2, 4

or

(2x + 4)(x + 4) = 0
x = -2, -4

FOIL out and we get:

2x² - 12x + 16 = 0
k = 12

or

2x² + 12x + 16 = 0
k = -12

k = ±12

Answer 3

9 nickles 40 5 pennies 4 dimes a million nickel 5 pennies 8 nickels 5 pennies 4 dimes a million quarter 2 dimes 2 dimes 5 nickels 10 pennies 7 nickels 10 pennies 3 dimes a million nickel 15 pennies 3 dimes 25 pennies 2 dimes 35 pennies a million dime 35 pennies 2 nickels 25 pennies 4 nickels 15 pennies 6 nickels heres some . . . . lol wish this facilitates!

Answer 4

The most no-brainer method for multiple-choice is to substitute all the answers into the equation and see which one fits.

Sorry no time to do this now, I need to get off the computer, just thought the method may help.

Answer 5

just use the Viete formulas for Sum and Product: for ax^2+bx+c=0
S= R + 2R = -b/(2a)
P= R*2R = c/a.................... just make the calculations for yourself ...

Answer 6

sum of da roots is -b/a
product is c/a

substitute n u'l get