a quadraticc equation ax^2+bx+c=0 has 4-√17 and 4+√17 as solutions. find the values of b and c if the vaule of a is 1?

If 4-√17 and 4+√17 are roots then (x-(4-√17)) and (x-(4+√17)) are factors of ax^2+bx+c.

a=1 so the expression ax^2+bx+c becomes x^2+bx+c

A quadratic expression is of the form ax^2+bx+c in which (x-r1)(x-r2) are factors and (x-r1)(x-r2) = x^2 -(r1+r2)x + r1r2 and r1 and r2 are roots of ax^2+bx+c=0.

c = -(4-√17) * -(4+√17) = 16-17 = -1

b = -(4-√17 + 4+√17) = -8

quadratic equation x^2+bx+c=0 has m ,n as solutions b = -(m+n) , c=m*n

ax^2 + bx + c = 0

[x - (4 - √17)][x - (4 + √17)] = 0

(x - 4 + √17)(x - 4 - √17) = 0

x^2 - 8x - 1 = 0

b = -8, c = -1

x^2 + bx + c = 0

has solutions 4±√17

a quadratic of this form with roots p,q

has b = -(p+q), c = p*q

b = -8, c = -1

given roots p,q then the quadratic factors (x-p)(x-q) = 0

x^2 -px-qx+pq = 0

x^2-(p+q)x+pq = 0

The quadratic equation has (4 - √17) as solutions, so you can factorize: [x - (4 - √17)]

The quadratic equation has (4 + √17) as solutions, so you can factorize: [x - (4 + √17)]

...and it gives us:

[x - (4 - √17)].[x - (4 + √17)] = 0

(x - 4 + √17).(x - 4 - √17) = 0

x² - 4x - x.√17 - 4x + 16 + 4√17 + x.√17 - 4√17 - 17 = 0

x² - 8x - 1 = 0 → then you can deduce that:

a = 1

b = - 8

c = - 1

A quadratic equation x^2+bx+c=0 has 4-√17 and 4+√17 as solutions, so it has (x - 4 + √17)(x - 4 + √17) = 0 as factors.

This is (√((x - 4)²) + √17)(√((x - 4)²) - √17) = 0 which is (x - 4)² - 17 = 0 (from the difference of two squares), yielding x² - 8x - 1 = 0

So b = -8 and c = -1

Recall that if you have an equation in factored form, you can use that to solve for the roots:

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

x = -2 and 1

So reversing this, if you subtract the roots from x, you can turn them back into factors. Your two roots are already conjugate pairs, so you don't have to worry about other conjugate roots.

Taking each root and subtracting them each from x, then multiplying them together and setting it equal to 0, we get:

[x - (4 - √17)][x - (4 + √17)] = 0

simplify:

(x - 4 + √17)(x - 4 - √17) = 0

x² - 4x - x√17 - 4x + 16 + 4√17 + x√17 - 4√17 - 17 = 0

If you did everything right, the irrational terms should all cancel out:

x² - 8x + 16 - 17 = 0

x² - 8x - 1 = 0

You are told that a = 1, and this is true here (if you wanted it something else you could multiply both sides by that value).

So the values of b and c are:

b = -8

c = -1

y = x^2 + bx + c

Or, in factored form:

y = (x − p) (x − q)

Distributing:

y = x^2 − (p + q) x + pq

Given p = 4−√17 and q = 4+√17:

y = x^2 − 8x + 16 − 17

y = x^2 − 8x − 1

http://www.wolframalpha.com/input/?i=ax%5E2%2Bbx%2...

Plug and chug after re-arranging.