Help. If p and q are the roots of the equation 2x^2-x-4=0. Find the equation whose roots are p-(q/p) and q-(p/q). Working please
I will use the sum and product of roots properties.
for the given :
sum of roots = p + q = 1/2
product of roots = pq = -4/2 = -2
new roots are p- q/p and q - p/q
sum of new roots = p - q/p + q - p/q
= p^2q + q^2p - q^2 - p^2
= pq(pq + qp) - ( (p+q)^2 - 2pq)
= -2(-4) - (1/4 + 4)
= 8 - 17/4
= 15/4
product of new roots
= (p- q/p)(q - p/q)
=pq - p^2/q - q^2/p + 1
= 1/2 + 1 - (p^3 + q^3)
aside:
(p+q)^3 = p^3 + 3p^2q + 3pq^2 + q^3
(p+q)^3 = p^3 + q^3 + 3pq( p + q)
p^3 + q^3 = (p+q)^3 - 3pq( p + q)
= 1/8 - 3(-2)(1/2)
back to
1/2 + 1 - (p^3 + q^3)
= 3/2 - (1/8 - 3(-2)(1/2) )
= 3/2 - (-23/8) = 35/8
new equation:
x^2 - 15x/4 + 35/8 = 0
times 8
8x^2 - 30x + 35 = 0
I think you better check my algebra and arithmetic
To find the equation whose roots are p - (q/p) and q - (p/q), let's start by finding the values of p and q.
Given the equation 2x^2 - x - 4 = 0, we can solve it to find the roots.
1. Start by using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a)
The coefficients of the equation are: a = 2, b = -1, and c = -4.
2. Substitute these values into the quadratic formula:
x = (-(-1) ± √((-1)^2 - 4 * 2 * (-4))) / (2 * 2)
= (1 ± √(1 + 32)) / 4
= (1 ± √33) / 4
So, the roots of the equation are p = (1 + √33) / 4 and q = (1 - √33) / 4.
Now, let's find the equation with roots p - (q/p) and q - (p/q).
1. The first root, p - (q/p), can be simplified as follows:
p - (q/p) = [(p * p) - q] / p
2. Replace p and q with their respective values:
[(1 + √33)/4 * (1 + √33)/4 - (1 - √33)/4] / [(1 + √33)/4]
= [(1 + √33)^2 - (1 - √33)] / (1 + √33)
= [1 + 2√33 + 33 - 1 + √33] / (1 + √33)
= (3 + 3√33) / (1 + √33)
3. Simplify by multiplying both the numerator and the denominator by (1 - √33):
= (3 + 3√33)(1 - √33) / (1 + √33)(1 - √33)
= (3 - 3√33 + 3√33 - 99) / (1 - 33)
= (-96) / (-32)
= 3
So, the first root is 3.
Now, let's find the second root, q - (p/q):
1. Substitute p and q values:
q - (p/q) = [(1 - √33)/4 - (1 + √33) / 4] / [(1 - √33)/4]
= [(1 - √33 - 1 - √33) / 4] / [(1 - √33)/4]
= (-2√33) / (1 - √33)
2. Simplify by multiplying both the numerator and the denominator by (1 + √33):
= (-2√33)(1 + √33) / (1 - √33)(1 + √33)
= (-2√33 - 66) / (1 - 33)
= (-2√33 - 66) / (-32)
= √33/16 + 33/16
= (√33 + 33) / 16
Therefore, the second root is (√33 + 33) / 16.
So, the equation whose roots are p - (q/p) and q - (p/q) is x^2 - (3x) + (√33 + 33)/16 = 0.
To find the equation whose roots are p - (q/p) and q - (p/q), we can use Vieta's formulas. Vieta's formulas relate the coefficients of a polynomial to its roots.
Let's start by finding the sum and product of the roots of the original equation 2x^2 - x - 4 = 0.
The sum of the roots (p + q) can be found by using the formula:
Sum of Roots = -b/a, where a and b are the coefficients of the quadratic equation. In this case, a = 2 and b = -1.
So, Sum of Roots = -(-1)/2 = 1/2.
The product of the roots (p * q) can be found by using the formula:
Product of Roots = c/a, where c is the constant term and a is the coefficient of x^2 in the quadratic equation. In this case, c = -4 and a = 2.
So, Product of Roots = -4/2 = -2.
Now, we can find the values of p - (q/p) and q - (p/q).
p - (q/p) = (p^2 - q)/p
q - (p/q) = (q^2 - p)/q
Substituting the values of p and q, we get:
p - (q/p) = (p^2 - q)/p = (p^2 - (-2))/p = (p^2 + 2)/p
q - (p/q) = (q^2 - p)/q = (q^2 - 1)/q
Now, we can use these values to form the desired equation.
The equation with roots p - (q/p) and q - (p/q) can be written as:
[(x - (p^2 + 2)/p)][(x - (q^2 - 1)/q)] = 0
Simplifying further, we can multiply through by pq:
[p(x - (p^2 + 2)/p)][q(x - (q^2 - 1)/q)] = 0
(xp - p^2 - 2)(xq - q^2 + 1) = 0
Expanding and rearranging terms, we get:
x^2(pq) - x[(p^2 + 2)q + (q^2 - 1)p] + (p^2 + 2)(q^2 - 1) = 0
So, the equation whose roots are p - (q/p) and q - (p/q) is:
x^2(pq) - x[(p^2 + 2)q + (q^2 - 1)p] + (p^2 + 2)(q^2 - 1) = 0