Question

If α,β are the roots of x^2+px+q=0, form quadratic equation whose roots are(α-β)^2 and (α+β)^2

Answer 1

Answer:

x² - (2p² - 4q)x + p²(p² - 4q) = 0

Note:

★ The possible values of the variable which satisfy the equation are called its roots or solutions.

★ A quadratic equation can have atmost two roots.

★ If α and ß are the roots of the quadratic equation ax² + bx + c = 0 , then ;

• Sum of roots , (α + ß) = -b/a

• Product of roots , (αß) = c/a

★ If α and ß are the roots of any quadratic equation , then it is given by ;

x² - (α + ß)x + αß = 0

★ If α and ß are the roots of the quadratic equation ax² + bx + c = 0 , then they (α and ß) are also the roots of the quadratic equation k(ax² + bx + c) = 0 , k ≠ 0.

Solution:

Here,

The given quadratic equation is ;

x² + px + q = 0 .

Clearly,

a = 1

b = p

c = q

Also,

It is given that ,

α and ß are the roots of the given quadratic equation.

Thus,

Sum of roots = -b/a

α + ß = -p/1 = -p

Also,

Product of roots = c/a

αß = q/1 = q

Now,

Let A and B be the roots of required quadratic equation .

Also,

It is given that , (α - ß)² and (α + ß)² are the roots of the required quadratic equation .

Thus,

=> A = (α - ß)²

=> A = (α + ß)² - 4αß

=> A = (-p)² - 4q

=> A = p² - 4q

Also,

=> B = (α + ß)²

=> B = (-p)²

=> B = p²

Now,

Sum of roots of the required quadratic equation will be ;

=> A + B = p² - 4q + p²

=> A + B = 2p² - 4q

Also,

Product of roots of the required quadratic equation will be ;

=> A•B = (p² - 4q)p²

=> A•B = p²(p² - 4q)

Thus,

The required quadratic equation will be ;

x² - (A + B)x + A•B

x² - (2p² - 4q)x + p²(p² - 4q) = 0