if p and q are the zeros of the quadratic polynomial 2x^2+2 (m+n)x+m^2+n^2form the quadratic polynomial whose zeros are (p+q)^2and (p-q)^2
Answers
Step-by-step explanation:
The given equation is :
2x² + 2(m+n)x + m² + n² =0 --------------------- 1
p and q are the zeros of the given equation.
General form of quadratic equation
ax² + bx + c =0 ------------------------------------ 2
Comparing equation 1 with 2
a= 2 , b = 2(m+n), c=m²+n²
We know that the roots of the quadratic equation is given by
Sum of roots = -b/c
Therefore sum of roots of given equation is:
p + q = -2(m+n)/2 = -(m+n)
Product of roots = c/a
Therefore, p.q = (m² +n²)/2
Required equations having roots as (p+q)² and (p-q)²
sum of roots,
(p+q)² + (p-q)² = (-(m+n))² + (p²+q²-2pq)
= (m+n)² + ((p+q)² - 2pq -2pq)
= (m+n)² +(m+n)² - 2(m²+n²)
= 2{m²+n²+2mn - m² - n²} =4mn
Product of roots,
(p+q)².(p-q)² = (m²+n² + 2mn)(m² + n² -2mn)
Therefore required equation is: x² - 4mnx + (m²+n²+2mn)(m²+n²-2mn)=0
Answer:
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