Question

If p and q are the zeroes of the quadratic polynomial 2x^2 + 2(m+n)x + m^2 + n^2 , form the quadratic polynomial whose zeroes are (p+q)^2 and (p-q)^2



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Answer 1

Use Vieta's method on $2x^2 + 2(m+n)x + m^2 + n^2$.

$\displaystyle\left \{ {{p + q=-(m + n)} \atop {pq =\dfrac{m^2 + n^2}{2}}} \right.$

Two zeros of the new polynomial:

$(p+q)^2=p^2+2pq+q^2$ and $(p-q)^2=p^2-2pq+q^2$

Construct the new polynomial with Vieta's method.

Sum and product of the new polynomial:

• Sum $2(p^2+q^2)$
• Product $(p+q)^2(p-q)^2$

Finding the sum:

$(p+q)^2-2pq=p^2+q^2$

$=(m+n)^2-(m^2+n^2)$

$=2mn$

→ $4mn$ is the sum.

Finding the product:

$(p+q)^2-4pq=(p-q)^2$

$=(m+n)^2-2(m^2+n^2)$

$=-(m^2-2mn+n^2)=-(m-n)^2$

→  $-(m+n)^2(m-n)^2$ is the product.

The new quadratic equation is $x^2-4mnx-(m+n)^2(m-n)^2$.

More information:

Vieta's Method

Consider a quadratic polynomial $x^2+\dfrac{b}{a} x+\dfrac{c}{a}$.

If α and β are the zeroes of the polynomial then

$(x-\alpha )(x-\beta )=x^2-(\alpha +\beta )x+\alpha \beta\;\textbf{[Factor Theorem]}$.

• $\alpha +\beta$ is the sum of the two zeroes.
• $\alpha \beta$ is the product of the two zeroes.

So $\alpha +\beta =-\dfrac{b}{a}$ and $\alpha \beta =\dfrac{c}{a}$.

Answer 2

Answer:

here it is

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