Question

in polynomial x^2+x-2 if a and b are zeros find 1/a-1/b​

Answer 1

EXPLANATION.

Quadratic equation.

⇒ x² + x - 2.

As we know that,

Sum of the zeroes of the quadratic equation.

⇒ α + β = -b/a.

⇒ α + β = -(1)/1 = -1.

Products of the zeroes of the quadratic equation.

⇒ αβ = c/a.

⇒ αβ = (-2)/1 = -2.

To find :

⇒ 1/α + 1/β.

⇒ β + α/αβ.

Put the values in the equation, we get.

⇒ -1/-2 = 1/2.

⇒ 1/α + 1/β = 1/2.

                                                                                                                         

MORE INFORMATION.

Nature of the factors of the quadratic expression.

(1) = Real and different, 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.

Answer 2

${\large{\pmb{\sf{\underline{Question...}}}}}$

${\red{\bigstar}}$ In a polynomial ${\red{\sf{x^{2} + x-2}}}$ , ${\red{\sf{\alpha}}}$ and ${\red{\sf{\beta}}}$ are the zeroes. Find out ${\red{\sf{\dfrac{1}{\alpha} - \dfrac{1}{\beta}}}}$

${\large{\pmb{\sf{\underline{Given \: that...}}}}}$

${\purple{\bigstar}}$ A polynomial is given as ${\red{\sf{x^{2} + x-2}}}$

${\purple{\bigstar}}$ ${\red{\sf{\alpha}}}$ and ${\red{\sf{\beta}}}$ are the zeroes of the given polynomial.

${\large{\pmb{\sf{\underline{To \: find...}}}}}$

${\purple{\bigstar}}$ The value of ${\red{\sf{\dfrac{1}{\alpha} - \dfrac{1}{\beta}}}}$ in the given polynomial.

${\large{\pmb{\sf{\underline{Solution...}}}}}$

${\purple{\bigstar}}$ The value of ${\red{\sf{\dfrac{1}{\alpha} - \dfrac{1}{\beta}}}}$ in the given polynomial = 3/-2

${\large{\pmb{\sf{\underline{Knowledge...}}}}}$

Some knowledge about Quadratic Equations -

★ Sum of zeros of any quadratic equation is given by ➝ α+β = -b/a

★ Product of zeros of any quadratic equation is given by ➝ αβ = c/a

★ A quadratic equation have 2 roots

★ ax² + bx + c = 0 is the general form of quadratic equation

⠀⠀⠀⠀━━━━━━━━━━━━━━━━━━

${\large{\pmb{\sf{\underline{Using \; concepts...}}}}}$

${\red{\bigstar}}$ Formula to find out the sum of zeros of any quadratic equation =

${\small{\underline{\boxed{\sf{\alpha + \beta \: = -b/a}}}}}$

${\red{\bigstar}}$ Formula to find out the product of zeros of any quadratic equation =

${\small{\underline{\boxed{\sf{\alpha \beta \: = c/a}}}}}$

${\large{\pmb{\sf{\underline{Full \; Solution...}}}}}$

~ Firstly we have to use the formula to find out the sum of zeros of any quadratic equation =

${\small{\underline{\boxed{\sf{\alpha + \beta \: = -b/a}}}}}$

${\sf{:\implies \alpha + \beta \: = -b/a}}$

${\sf{:\implies Given \: = x^{2} + x-2}}$

${\sf{:\implies \alpha + \beta \: = -(1)/1}}$

${\sf{:\implies \alpha + \beta \: = -1/1}}$

${\sf{:\implies \alpha + \beta \: = -1}}$

~ Now we have to use the formula to find out the product of zeros of any quadratic equation =

${\small{\underline{\boxed{\sf{\alpha \beta \: = c/a}}}}}$

${\sf{:\implies \alpha \beta \: = c/a}}$

${\sf{:\implies \alpha \beta \: = c/a}}$

${\sf{:\implies Given \: = x^{2} + x-2}}$

${\sf{:\implies \alpha \beta \: = -2/1}}$

${\sf{:\implies \alpha \beta \: = -2}}$

~ Now we have to use substitution method we get the following result,

${\sf{:\implies By \: substituting \: values \: we \: get}}$

${\sf{:\implies x^{2} + x-2}}$

${\sf{:\implies x^{2} + x-2 = 0}}$

${\sf{:\implies x^{2} + 2x - x - 2 = 0}}$

${\sf{:\implies x(x+2) -1(x+2) = 0}}$

${\sf{:\implies x \: = -2 \: or \: x \: = -1}}$

~ Now we have to use an identity and have to find out the value of (β-α)

${\small{\underline{\boxed{\sf{(a-b)^{2} \: = (a+b)^{2} - 4ab}}}}}$

${\sf{:\implies (a-b)^{2} \: = (a+b)^{2} - 4ab}}$

${\sf{:\implies (a-b)^{2} \: = (-1)^{2} - 4(-2)}}$

${\sf{:\implies (a-b)^{2} \: = 1 + 8}}$

${\sf{:\implies (a-b)^{2} \: = 9}}$

${\sf{:\implies (a-b) = \sqrt{9}}}$

${\sf{:\implies (a-b) = 3}}$

${\sf{:\implies (b-a) = -3}}$

${\sf{:\implies Therefore, \: (\beta - \alpha) \: = -3}}$

~ Now finding the value of 1/α - 1/β

${\sf{:\implies \dfrac{1}{\alpha} - \dfrac{1}{\beta}}}$

${\sf{:\implies (b-a)/ab}}$

${\sf{:\implies 3/-2}}$

Henceforth, the value of 1/α - 1/β = 3/-2