If f(x) = 4x ² - 16x + m = 0 , if one of roots 3 more the other root , find m & roots?

Question

It is not my homework , i am just trying to check my answer , if i got corect or not .

Thanks Friends

Answer 1

Well dear

-If the first root is " α " and the second roots is " α + 3" so have
▪x ² - (α + β) + α β = 0
▪( x - α )(x - β ) = 0
-b/a = α+ β
&
c/a = α β
so we have ;
c/a = m/4
-b/a = - ( -16) / 4 = 4
α + α + 3 = 4 , 2α + 3 = 4
α = 1/2 , β = 3 1/2
2 /a = α β => m/4 = 1/2 * 7/2 , m = 7

Good Luck dear

Answer 2

For polynomials you must remember that if the leading coefficient is 1, then the second leading coefficient is always the negative of the sum of ALL the roots of the polynomial and the constant at the end is always the product of ALL the roots.

so first divide everything by 4.
x^2-4x+m/4=0

Now you know that both of the roots have to add up to 4 (not negative four, you must change the sign).

So A is one of the roots, A+3 has to be the other. which means that
A+A+3=4 => A=1/2 => A+3=7/2

Now we also know that A(A+3)=m/4=(1/2)(7/2) which gives you that m=7.

Everyone aboveme is basically doing the same thing, except that it canbe done generally for any polynomial so just remember the result. The sum of all of the roots (for any degree polynomial with the leading coefficient one) is negative of the second number and the product of all the roots is always the constant.

Answer 3

The following information fits the requirements of your question.
Given:
f(x) = 4x ² - 16x + m = 0
One of the roots is 3 (given), so it must satisfy the about equation.
Let's substitute it in.
4x ² - 16x + m = 0
4(3)² - 16(3) + m = 0
36 - 48 + m = 0
m = 12

So the equation is:
4x ² - 16x + 12 = 0
Using quadratic formula:
[-b ± √(b² - 4ac)] / 2a
[-(-16) ± √((-16)² - 4(4)(12))] / 2(4)
[16 ± √(256 - 192)] / 8
[16 ± √(64)] / 8
16 ± 8 / 8
So x = 3 (or) x =1.

Answer 4

Your roots are A and A+3.

Divide your original equation by 4 (this will not affect the roots), so that you can get x ² - 4x + m/4 = 0.

Factoring this, we get:

x ² - 4x + m/4 = (x-A)(x-(A+3))

Now using FOIL we know that:

A(A+3) = m/4
A + A + 3 = 4

Solving for A we get 2A = 1, or A=1/2

So our two factors are 1/2 and 7/2

and m = (1/2 * 7/2) = 7

- - -

Check the answers by plugging into your original equation:

4x² - 16x + 7 = 0

(2x - 7) (2x - 1) = 0

x = 7/2 or x = 1/2

Therefore, it checks out!

~ ♥ ~

Answer 5

if (x-a)(x-b) = 0 then x^2 - (a+b)x + ab = 0

4x^2 -16x + m = 0 means x^2 -4 +m/4 = 0

so (a+b) = 4
if (a-b) = 3
then 2a = 7 so a = 7/2
and 2b = 1 so b = 1/2


try it
(x - 7/2)(x - 1/2) = 0
means
(2x - 7)(2x -1 ) = 0
means
4x^2 -16x +7 = 0


so m = 7


also we could have argued that

m/4 = a b
and so m would have to be 7

product of solutions a and b = (coeff of x^0)/(coefficient of x^2)
therefore ab = m/4

or by comparing the two equations

x^2 - (a+b)x + ab = 0
x^2 -4 +m/4 = 0