Question

If alpha and beta are the zeros of the quadratic polynomial f(x) = x^2 -7x +P, such that alpha ^2+ beta^2 = 29 . Find value of P.

Answer 1

$\huge \green{\mathfrak{solution}}$

Given:

$\alpha \: and \: \beta \: are \: the \: zeroes \:$

$f(x) = {x}^{2} - 7x + p$

and

${ \alpha }^{2} + { \beta }^{2} = 29$

To find :

P = ?

Here ,

a = 1 , b=-7 and c= p

$\alpha + \beta = \frac{ - b}{a} = \frac{ - ( - 7)}{1} =7$

and

$\alpha \beta = \frac{c}{a} = \frac{p}{1} = p$

Now ,

$= > { \alpha }^{2} + { \beta }^{2} = 29 \\ \\ = > ( { \alpha + \beta )}^{2} - 2 \alpha \beta = 29 \\ \\ = > ( {7)}^{2} - 2p = 29 \\ \\ = > 49 - 2p = 29 \\ \\ = > 49 - 29 = 2p \\ \\ = > 20 = 2p \\ \\ = > p = \cancel \frac{20}{2} \\ \\ = > p = 10$

Hence, the value of P is 10

Answer 2

Question :-- If alpha and beta are the zeros of the quadratic polynomial f(x) = x^2 -7x +P, such that alpha ^2+ beta^2 = 29 . Find value of P.

Formula used :---

→ The sum of the roots of the Equation ax² + bx + c = 0 , is given by = (-b/a)

and ,

→ Product of roots of the Equation is given by = c/a.

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solution :--

Equation x²-7x + p = 0 have ,

→ a = 1

→ b = (-7)

→ c = p

→ it has been given that roots of the Equation x^2 -7x +P = 0 are α and β ..

As told above ,

→ sum of roots = -b/a = -(-7)/1= 7

→ product of roots = c/a = p/1 = p

______________________

Now,

it has been given that, α² + β² = 29

or,

=> (α + β)² = α² + β² + 2 α*β

Putting values now we get,

→ 7² = 29 + 2 * p

→ 49 = 29 + 2p

→ 2p = 49 - 29

→ 2p = 20

Dividing both sides by 2 now we get,

→ p = 10

______________________

Hence, we can say that the value of p will be 10..