Question

if alpha and beta are the zeroes of quadratic equation p(x) 2x2-kx+ 5 and alpha +beta whole square -1/2 and alpha into beta =23/2 find the value of k​

Answer 1

Answer: k = ± 6

Given that, α & β are the zeros of the quadratic equation:

• p(x) = 2x² - kx + 5

And, ( α + β )² + αβ = (23/2) [Corrected Question]

We are required to find the value of 'k'

Concept used in this question is:

$\implies \text{Sum of zeros of p(x)} = \dfrac{ -(\text{coefficient of x term})}{\text{coefficient of}\:x^2\:\text{term}}$

$\implies \text{Product of zeros of p(x)} = \dfrac{ -(\text{constant term})}{\text{coefficient of}\;x^2\:\text{term}}$

According to the question,

$\implies \text{Sum of zeros} = \alpha + \beta = \dfrac{-(-k)}{2}\\\\\\\implies \boxed{( \alpha + \beta ) = \dfrac{k}{2}}$

Similarly,

$\implies \boxed{\text{Product of zeros} = \alpha\beta = \dfrac{5}{2}}$

Using the given information we get:

$\implies (\alpha + \beta)^2 + ( \alpha\beta) = \dfrac{23}{2}\\\\\\\text{Substituting the respective terms we get:}\\\\\\\implies (\dfrac{k}{2})^2 + \dfrac{5}{2} = \dfrac{23}{2}\\\\\\\implies \dfrac{k^2}{4} + \dfrac{5}{2} = \dfrac{23}{2}\\\\\\\implies \dfrac{k^2}{4} = \dfrac{23}{2} - \dfrac{5}{2}\\\\\\\implies \dfrac{k^2}{4} = \dfrac{18}{2} = 9\\\\\\\implies k^2 = 9 \times 4 = 36\\\\\\\implies \boxed{k = \sqrt{36} = \pm 6}$

Hence the value of 'k' can be 6 or -6.

Answer 2

Question:

$p(x) = 2 {x}^{2} - kx + 5$

Given that :

$( { \alpha + \beta )}^{2} + \alpha \beta = \frac{23}{2}$

Find the value of k

Answer:

General equation of quadratic equation =

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

we know that,

$( \alpha + \beta ) = \frac{ - b}{a} \: \: and \: \: \alpha \beta = \frac{c}{a}$

$\alpha + \beta = \frac{ - co \: efficient \: of \: x}{co \: efficient \: of {x}^{2} }$

$\alpha + \beta = \frac{ - ( - k)}{2}$

$\alpha + \beta = \frac{k}{5}$

$\alpha \beta = \frac{constant \:}{co \: efficient \: of \: {x}^{2} }$

$\alpha \beta = \frac{5}{2}$

Since,

(k/2)^2 + (5/2) = 23/2

k^2/4 + 5/2 = 23/2

k^2/4 = 23/2 -5/2

k^2/4 = 18/2

k^2 = 9×4

k^2 = 36

k = + (or) - 6