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harkovskaia [24]
3 years ago
5

If alpha and beta are the zeroes of quadratic polynomial ax^2+bx+c , then find alpha^4+beta^4

Mathematics
1 answer:
weqwewe [10]3 years ago
5 0
Ill write A for alpha and B for beta.

AB = c/a  and A + B = -b/a

A^4 + B^4 = (A^2 + B^2)^2  -   2A^2B^2

                 = [(A + B)^2 - 2AB] ^2 - 2A^2B^2

Plugging in the values for A+B and AB we get

A^4 + B^4 = [(-b/a)^2 - 2c/a]^2 - 2(c/a)^2

   =    (b^2 / a^2 - 2c / a)^2 - 2c^2/a^2

  = (b^2 - 2ac)^2  - 2c^2
     ----------------      -----
          a^4                a^2

 =   (b^2 - 2ac)^2 - 2a^2c^2
       -----------------------------
                   a^4
            
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8 0
3 years ago
Read 2 more answers
Consider the quadratic equation ax^2+bx+5=0,where a, b and c are rational numbers and the quadratic has two distinct zeros.
FrozenT [24]
You have shared the situation (problem), except for the directions:  What are you supposed to do here?  I can only make a educated guesses.  See below:

Note that if    <span>ax^2+bx+5=0    then it appears that c = 5 (a rational number).

Note that for simplicity's sake, we need to assume that the "two distinct zeros" are real numbers, not imaginary or complex numbers.  If this is the case, then the discriminant,    b^2 - 4(a)(c), must be positive.  Since c = 5, 

b^2 - 4(a)(5) > 0, or b^2 - 20a > 0.

Note that if the quadratic has two distinct zeros, which we'll call "d" and "e," then 

(x-d) and (x-e) are factors of ax^2 + bx + 5 = 0, and that because of this fact,

         - b plus sqrt( b^2 - 20a )
d =  ------------------------------------
                      2a

and

 </span>         - b minus sqrt( b^2 - 20a )
e =  ------------------------------------
                      2a

Some (or perhaps all) of these facts may help us find the values of "a" and "b."  Before going into that, however, I'm asking you to share the rest of the problem statement.  What, specificallyi, were you asked to do here?

7 0
3 years ago
Question is in picture.
Norma-Jean [14]

Answer:

<h2>B. 2x + y = 4</h2>

Step-by-step explanation:

Having the system of equations in its simplest form

\left\{\begin{array}{ccc}ax+by=c\\dx+ey=f\end{array}\right

If

a=d,\ b=e,\ c=f

then the system of equations has infinitely many solutions.

If

a=d,\ b=e,\ c\neq f

then the system of equations has no solution.

If

a\neq d\ or\ b\neq e

then the system of equations has one solution.

We have the equation

y=-2x+4

Convert to the standard form Ax + By = C<em>:</em>

<em />y=-2x+4<em>              add 2x to both sides</em>

y+2x=-2x+2x+4\\\\2x+y=4

3 0
3 years ago
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