Find a quadratic polynomial whose roots are 3+ root 3 and 3- root 3
Answers
EXPLANATION.
Quadratic polynomial.
Whose roots = (3 + √3) and (3 - √3).
As we know that,
Sum of the zeroes of the quadratic equation.
⇒ α + β = -b/a.
⇒ (3 + √3) + (3 - √3).
⇒ 3 + √3 + 3 - √3.
⇒ 6.
⇒ α + β = 6.
Products of the zeroes of the quadratic equation.
⇒ αβ = c/a.
⇒ (3 + √3)(3 - √3).
As we know that,
Formula of :
⇒ (x² - y²) = (x + y)(x - y).
Using this formula in equation, we get.
⇒ [(3)² - (√3)²].
⇒ [9 - 3] = 6.
⇒ αβ = 6.
As we know that,
Formula of quadratic polynomial.
⇒ x² - (α + β)x + αβ.
Put the values in the equation, we get.
⇒ x² - (6)x + 6 = 0.
⇒ x² - 6x + 6 = 0.
MORE INFORMATION.
Nature of the roots of the quadratic expression.
(1) = Real and unequal, if b² - 4ac > 0.
(2) = Rational and different, if b² - 4ac is a perfect square.
(3) = Real and equal, if b² - 4ac = 0.
(4) = If D < 0 Roots are imaginary and unequal Or complex conjugate.