Question

If α, β are the roots of the quadratic equation 2x2 + 5x + k then the value of k if
α2 +β2 +αβ = 21 4

Answer 1

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Q) If $\alpha$ and $\beta$ are the roots of the Quadratic Equation 2x² + 5x + k .

Then the value of k if

${ \alpha }^{2} + { \beta }^{2} + \alpha \beta = \frac{21}{4}$

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✷ Concept :

• In these type of Questions , we Simply use the sum and product of roots of Quadratic Equation .

• In this question , we would find the sum and product of the Equation.

• Then we will square the sum and after simplifying , we will get the value of k .

• Standard Quadratic Equation is in the form :

$\sf \: a {x}^{2} + bx + c = 0$

Where ,

• a ≠ 0

✷ Given :

• $\alpha$ and $\beta$ are the zeroes of the Equation .
• Equation = 2x² + 5x + k
• ${ \alpha }^{2} + { \beta }^{2} + \alpha \beta = \frac{21}{4}$

✷ To Find :

• The value of k

✷ Solution :

$2 {x}^{2} + 5x + k = 0$

here ,

• a = 2
• b = 5
• c = k

So ,

$\rm \: sum \: of \: zeroes = ( \alpha + \beta ) = \frac{ - b}{a}$

$\mapsto \boxed{\rm \: \alpha + \beta = \frac{ - 5}{2} }.....(i)$

Now ,

$\rm \: product \: of \: zeroes = \alpha \beta = \frac{c}{a}$

$\mapsto \boxed{ \rm \: \alpha \beta = \frac{k}{2} }......(ii)$

Now ,

Squaring both sides of eq (i)

$\mapsto \rm \: {( \alpha + \beta )}^{2} = {( \frac{ - 5}{2}) }^{2}$

$\mapsto \rm \: { \alpha }^{2} + { \beta }^{2} + 2 \alpha \beta = \frac{25}{4}$

$\mapsto \rm \: \underline{ { \alpha }^{2} + { \beta }^{2} + \alpha \beta} + \alpha \beta = \frac{25}{4}$

Since , we know that

${ \alpha }^{2} + { \beta }^{2} + \alpha \beta = \frac{21}{4}$

so , Putting values..

$\mapsto \rm \: \frac{21}{4} + \frac{k}{2} = \frac{25}{4}$

$\mapsto \rm \: \frac{k}{2} = \frac{25}{4} - \frac{21}{4}$

$\mapsto \rm \: \frac{k}{2} = \frac{25 - 21}{4}$

$\mapsto \rm \: \frac{k}{2} = \frac{ \cancel4}{ \cancel4}$

$\mapsto \rm \: \frac{k}{2} = 1$

$\mapsto \boxed{ \rm \: k = 2}$

So,

The value of k is 2

Answer 2

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