If a + B = -2 and a3 + b3 = 28, then thequadratic equation having roots a and Bwill be....
Answers
EXPLANATION.
⇒ a + b = - 2.
⇒ a³ + b³ = 28.
As we know that,
We can write equation as,
⇒ a³ + b³ = 28.
As we know that,
Formula of :
⇒ x³ + y³ = (x + y)(x² + y² - xy).
Using this formula in the equation, we get.
⇒ (a + b)(a² + b² - ab) = 28.
Put the values of a + b = - 2 in the equation, we get.
⇒ (-2)(a² + b² - ab) = 28.
⇒ (a² + b² - ab) = - 14.
As we know that,
Formula of :
⇒ x² + y² = (x + y)² - 2xy.
Using this formula in the equation, we get.
⇒ [(a + b)² - 2ab - ab] = - 14.
⇒ [(a + b)² - 3ab] = - 14.
Put the value of a + b = - 2 in the equation, we get.
⇒ [(-2)² - 3ab] = - 14.
⇒ [4 - 3ab] = - 14.
⇒ 4 - 3ab = - 14.
⇒ 4 + 14 = 3ab.
⇒ 18 = 3ab.
⇒ ab = 6.
Sum of the zeroes of the quadratic polynomial.
⇒ a + b = - b/a.
⇒ a + b = - 2. - - - - - (1).
Products of the zeroes of the quadratic polynomial.
⇒ ab = c/a.
⇒ ab = 6. - - - - - (2).
As we know that,
Formula of quadratic polynomial.
⇒ x² - (a + b)x + ab.
Put the values in the equation, we get.
⇒ x² - (-2)x + 6.
⇒ x² + 2x + 6.
MORE INFORMATION.
Range of quadratic equation.
(1) = For y = ax² + bx + c, if a > 0.
⇒ F(x) ∈ [-D/4a, ∞).
(2) = For y = ax² + bx + c, if a < 0.
⇒ F(x) ∈ (-∞, -D/4a].
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