Question

if alpha and beta are the zeros of x square - 6 X + K what is the value of k if 3 alpha + 2 B tech 20.
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Answer 1

Answer

• k = - 16

Given

• α , β are the zeroes of x² - 6x + k

• 3α + 2β = 20

To Find

• Value of k

Solution

Compare given polynomial x² - 6x + k with ax² + bx + c , we get ,

• a = 1 , b = - 6 , c = k

3α + 2β = 20 ... (1) [ Given ]

Also given that α , β are zeroes of the polynomial .

So ,

Sum of zeroes ,

α + β = -b/a = -(-6)/1

⇒ α + β = 6 ... (2)

Product of zeroes ,

αβ = c/a = k/1

⇒ αβ = k ... (3)

On solving (1) & (2) , i.e., (1) - 2×(2) , we get ,

⇒ ( 3α + 2β ) - 2 ( α + β ) = 20 - 2(6)

⇒ 3α + 2β - 2α - 2β = 20 - 12

⇒ α = 8

On sub. α value in (2) , we get ,

⇒ 8 + β = 6

⇒ β = 6 - 8

⇒ β = - 2

On sub. α , β values in (3) , we get ,

⇒ (8)(-2) = k

⇒ - 16 = k

⇒ k = - 16

More Info

$\boxed{\begin{minipage}{6.6cm} \rm For\ a\ cubic\ polynomial\ ax^3+bx^2+cx+d ,\\\\\rm Sum\ of\ zeroes,\\\\\rm {\star \; \alpha+\beta+\gamma=\dfrac{-b}{a} }\\\\\rm Sum\ of\ Product\ of\ zeroes,\\\\\rm \star \; \alpha\beta+\beta\gamma+\gamma\alpha=\dfrac{c}{a}\\\\\rm Product\ of\ zeroes,\\\\\star \; \alpha \beta \gamma=\dfrac{-d}{a} \end{minipage}}$

Answer 2

$\rm\large\green{\underline{ Question : }}$

If α and ß are the zeroes of the polynomial

p(x) = x² - 6x + k. What is the value of 3α + 2ß = 20.

$\rm\large\green{\underline{ Solution : }}$

★ Given that,

• P(x) = x² - 6x + k
• 3α + 2ß = 20 ..... 1

★ To find,

• The value of k.

★ Let,

• a = 1
• b = - 6
• c = k

We know that,

$\tt\green{\implies Sum\:of\:the\:zeroes\:= \alpha + \beta = \frac{-b}{a}}$

• Substitute the values

$\sf\:\implies \alpha + \beta = \frac{-(-6)}{1}$

$\sf\:\implies \alpha + \beta = 6 ..... 2$

$\tt\green{ \implies Product\:of\:the\:zeroes\:= \alpha\beta = \frac{c}{a}}$

• Substitute the values.

$\sf\:\implies \alpha\beta = \frac{k}{1}$

$\sf\:\implies \alpha\beta = k .....3$

★ Now,

Multiply equation 1 with 2. We get,

$\sf\:\implies 2\alpha + 2\beta = 12 .....4$

From the equations 1 & 4,we get,

$\sf\:\implies - \alpha = -8$

$\sf\:\implies \alpha = 8$

$\boxed{ \alpha = 8 }$

Substitute the value of α in 1,to get ß.

$\sf\:\implies 8 + \beta = 6$

$\sf\:\implies \beta = 6 - 8$

$\sf\:\implies \beta = - 2$

$\boxed{ \beta = - 2 }$

Now, substitute the values of α & ß in 3,to get k.

$\sf\:\implies 8(-2) = k$

$\sf\:\implies k = - 16$

$\underline{\boxed{\bf{\orange{ \therefore The\:value\:of\: k \: is\: - 16}}}}\:\red{\bigstar}$