Question

If
$\alpha$
and
$\beta$
are zeroes of polynomial 3x² -8x -3, then find the value of (
$(\alpha + \beta) ^{2} - 2 \alpha \beta$

​

Answer 1

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

★ If α(alpha) and β(beta) are the zeroes of polynomial 3x²-8x-3, then find the value of (α+β)² - 2αβ.

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

★ Polynomial = 3x²-8x-3

★ α and β are the zeroes of polynomial

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

★ The value of (α+β)² - 2αβ

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

★ The value of (α+β)² - 2αβ = 46/9

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

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

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

~ As we already provided with a polynomial as 3x²-8x-3 and it is said that α and β are the zeroes of polynomial and we are asked to find the value of (α+β)² - 2αβ. So firstly we have to find out the value for α+β and αβ by using the given formulas:

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

★ 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

$\: \: \: \: \: \: \: \: \sf Here \begin{cases} & \sf{b \: is \: \bf{8}} \\ & \sf{a \: is \: \bf{3}} \\ & \sf{c \: is \: \bf{3}} \end{cases}$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀Means,

⠀⠀⠀⠀● α+β = -8/3

⠀⠀⠀⠀● αβ = 3/3 = 1

~ Now finding the value of (α+β)² - 2αβ by using the above finded values. We just have to put the values and have to solve. Let us solve this question!

${\sf{\leadsto (\alpha + \beta)^{2} - 2 \alpha \beta}}$

${\sf{\leadsto (-8/3)^{2} - 2(1)}}$

${\sf{\leadsto -8/3 \times -8/3 - 2 \times 1}}$

${\sf{\leadsto -8/3 \times -8/3 \times 2}}$

${\sf{\leadsto 64/9 - 2/1}}$

${\sf{\leadsto 46/9}}$

• Henceforth, the value of (α+β)² - 2αβ is 46/9

Answer 2

Let me answer it...

See the above attachment for answer.