(1)given that -2x^2+6x+7=0, if alfa and B are the roots of the equation, form new equation whose roots are(a)alfa/B and B/alfa (b)alfa/B^2and B/alfa^2(b) if alfa and B are the roots of the equation 4x^+8x-1=0. find the values of alfa^3 B+alfaB^3
x^2 - 3 x - 7/2 = 0
x = [ 3 +/-sqrt (9+14) ]/2
A = [ 3 + 23]/2 = 13
B = [ 3 - 23] / 2 = -10
(a)
(x- A/B)(x-B/A) = 0
(x +13/10)(x+10/13) = 0
x^2 + (269/130) x + 1 = 0
130 x^2 + 269 x + 130 = 0
etc etc etc
Or
using a property of quadratic roots:
for ax^2 + bx + c = 0
if p and q are the roots
p+q = -b/a
qp = c/a
so for the given equation:
p+q = -6/-2 = 3
pq = -7/2
a)
for new equation:
roots are p/q and q/p
sum = p/q + q/p
= (p^2+q^2)/(pq)
= ( (p+q)^2 - 2pq)/(pq)
= (9 - 2(-7/2))/(-7/2)
= 16(-2/7)
= -32/7
product = (p/q)(q/p) = 1 = 7/7
so the equation is 7x^2 + 32x + 7 = 0
( Damon's roots should have been:
A = [ 3 + √23]/2
B = [ 3 - √23] / 2 )
b) root of new equation are p/q^2 and q/p^2
new sum = p/q^2 + q/p^2
= (p^3 + q^3)/(pq)^2
tricky part here ...
(p + q)^3 = p^3 + 3p^2q + 3pq^2 + q^3
= p^3 + q^3 + 3pq(p+q)
3^3 = p^3 + q^3 + 3(-7/2)(3)
27 + 63/2 = p^3 + q^3
p^3 + q^3 = 117/2
new sum = (117/2) / (49/4)
= (117/2)(4/49)
= 234/49
new product = (p/q^2)(q/p^2)
= pq/(pq)2
= 1/pq
= -2/7
new equation:
x^2 - (234/49)x - 2/7 = 0
times 49
49x^2 - 234x - 14 = 0
. Using x^2 +(p+q)x +pq new equatin is x^2 -3p -7/2 =2x^2 -6x - 7. . But ur ques is nt clear
To solve this problem, we will use Vieta's formulas, which relate the coefficients of a polynomial to its roots.
Given the equation -2x^2 + 6x + 7 = 0, we can start by finding the sum and product of the roots using Vieta's formulas:
The sum of the roots (α + B) is given by the formula: -(sum of coefficients of x) / leading coefficient.
In our case, the leading coefficient is -2 and the sum of the coefficients of x is 6.
So, (α + B) = -6 / -2 = 3.
The product of the roots (α * B) is given by the formula: constant term / leading coefficient.
In our case, the constant term is 7, and the leading coefficient is -2.
So, (α * B) = 7 / -2 = -3.5.
Now, let's answer the questions:
(a) Form new equation whose roots are α/B and B/α:
We can use the fact that if α and B are the roots of a quadratic equation Ax^2 + Bx + C = 0, then x = α/B and x = B/α will be the roots of the equation Bx^2 + Cx + A = 0.
In our case, with (α + B) = 3 and (α * B) = -3.5, the new equation with roots α/B and B/α will be Bx^2 + 3x - 3.5 = 0.
(b) Form new equation whose roots are α/B^2 and B/α^2:
Similar to the previous case, the new equation will be B^2x^2 + 3x - 3.5 = 0.
Now, let's move on to the second part of the question:
Given the equation 4x^2 + 8x - 1 = 0, we can again use Vieta's formulas to find the values of α and B.
The sum of the roots (α + B) is -(sum of coefficients of x) / leading coefficient.
In our case, the leading coefficient is 4, and the sum of the coefficients of x is 8.
So, (α + B) = -8 / 4 = -2.
The product of the roots (α * B) is the constant term / leading coefficient.
In our case, the constant term is -1, and the leading coefficient is 4.
So, (α * B) = -1 / 4 = -0.25.
Now, we can calculate the values of α^3 B + α B^3:
(α^3 B + α B^3) = (αB) (α^2 + B^2).
From Vieta's formulas, we know that:
(α^2 + B^2) = (α + B)^2 - 2(α * B)
We already found (α + B) = -2 and (α * B) = -0.25.
Substituting these values, we get:
(α^2 + B^2) = (-2)^2 - 2(-0.25) = 4 + 0.5 = 4.5.
And finally, plugging (α^2 + B^2) = 4.5 and (α * B) = -0.25 into the equation, we have:
(α^3 B + α B^3) = (-0.25) * (4.5) = -1.125.
Therefore, the value of α^3 B + α B^3 is -1.125.