# maths

posted by .

Given that a and b are the roots of the quadratic equation 2x^2 - 4x +3 = 0, find the quadratic equation in x with roots
a^2-(1/b) and b^2-(1/a).

• maths -

We can further transform the given equation in such a way that the numerical coefficient of x^2 is 1. That is, by dividing all terms by 2. In this form, recall that the sum of roots is equal to negative of the numerical coeff of x, and the product of roots is equal to the constant:
2x^2 - 4x + 3 = 0
x^2 - 2x + 3/2 = 0
Thus, if a and b are the roots,
a + b = 2
ab = 3/2
Given the roots of the quadratic equation we're looking for, we can also get their sum & products:
sum: a^2-(1/b) + b^2-(1/a) = [(ab)(a^2 + b^2) - (a+b)]/ab
product: (a^2-(1/b))*(b^2-(1/a)) = (a^2)(b^2) + 1/ab - (a+b)
From the values of a + b and ab, we can actually use them in order to get the values of the long expressions above.
Squaring the sum equation:
a + b = 2
a^2 + 2ab + b^2 = 4
Since we only need a^2 + b^2, we subtract 2ab to both sides:
(a^2 + 2ab + b^2) - 2ab = 4 - 2ab
a^2 + b^2 = 4 - 2(3/2)
a^2 + b^2 = 1
Substituting,
sum:
[(ab)(a^2 + b^2) - (a+b)]/ab
[(3/2)(1) - 2]/(3/2)
(3/2 - 2)*2/3
= -1/3

product:
(a^2)(b^2) + 1/ab - (a+b)
(3/2)^2 + 1/3/2 - (2)
9/4 + 2/3 - 2
= 11/12

Therefore, we can now write the equation:
x^2 - (sum)x + (product) = 0
x^2 + 1/3 x + 11/12 = 0

The other way to do this is to find the exact roots of the first equation then substitute it to the long expressions. But for me I think that is harder if the roots contain imaginary terms.

Hope this helps~ :)

## Similar Questions

1. ### Algebra2

Find the polynomials roots to each of the following problems: #1) x^2+3x+1 #2) x^2+4x+3=0 #3) -2x^2+4x-5 #3 is not an equation. Dod you omit "= 0" at the end?
2. ### Precalculus

"Show that x^6 - 7x^3 - 8 = 0 has a quadratic form. Then find the two real roots and the four imaginary roots of this equation." I used synthetic division to get the real roots 2 and -1, but I can't figure out how to get the imaginary …
3. ### Algebra

Given the roots, 1/2 and 4, find: (A) The quadratic equation 2x^2-9x+4 ?
4. ### Algebra

Find the number of integer quadruples (a,b,c,d) with 0\leq a,b,c,d \leq 100, such that a and b are the roots of the quadratic equation x^2-cx+d=0, while c and d are the roots of the quadratic equation x^2-ax+b.
5. ### Algebra

Find the number of integer quadruples (a,b,c,d) with 0\leq a,b,c,d \leq 100, such that a and b are the roots of the quadratic equation x^2-cx+d=0, while c and d are the roots of the quadratic equation x^2-ax+b
6. ### Math (Algebra)

If the ratio of the roots of the quadratic equation x^2−28x+A=0 is 2:5, what is the product of the roots of the quadratic equation x^2+Ax+2A+3=0?