Question

if alpha, beta re the zeros of the quadratic polynomial 2x2-4x+5 find the value of (i) alpha2 + beta2 (ii) (alpha-beta)2​

Answer 1

Question:

If α , β are the zeroes of the quadratic polynomial 2x²- 4x + 5, find the value of

(i) α² + β²

(ii) (α - β)²

To find:

$To \: find \: the \: value \: of:$

$1) \: { \alpha }^{2} + { \beta }^{2}$

$2) \: {( \alpha - \beta )}^{2}$

Answer:

Given:

An quadratic equation is given,

$2 {x}^{2} - 4x + 5 = 0$

$(i) { \alpha }^{2} + { \beta }^{2}$

$(ii) {( \alpha - \beta )}^{2}$

Step-by-step explanation:

$2 {x}^{2} - 4x + 5 = 0$

$It's \: of \: the \: form,$

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

a = 2 , b = -4 , c = 5

$\boxed{Sum \: of \: roots = \frac{ - b}{a} }$

$\alpha + \beta = \frac{ - ( - 4)}{2}$

$\boxed{ \alpha + \beta = 2}$

$\boxed{Product \: of \: roots = \frac{ c}{a} }$

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

$\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

$1) \: { \alpha }^{2} + { \beta }^{2}$

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

Now, substituting the values

$\implies {(2)}^{2} - \not{2}\big(\frac{5}{ \not{2}}\big)$

$\implies 4 - 5$

$\implies \boxed{ { \alpha }^{2} + { \beta }^{2} = - 1}$

$\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

$2) \: {( \alpha - \beta )}^{2}$

$\implies \underline{ \bold{ { \alpha }^{2} + { \beta }^{2} } }- 2 \alpha \beta$

$\implies {( - 1)}^{2} - \not{2} \big( \frac{5}{ \not{2}} \big)$

$\implies 1 + 5$

$\implies \boxed{ {( \alpha - \beta )}^{2} = 6}$

$\underline{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: }$

Formula used:

$\bigstar Sum \: of \: roots = \frac{ - b}{a}$

$\bigstar Product \: of \: roots = \frac{ c}{a}$

$\bigstar { \alpha }^{2} + { \beta }^{2} = {( \alpha + \beta )}^{2} - 2 \alpha \beta$

$\bigstar {( \alpha - \beta )}^{2} = { \alpha }^{2} + { \beta }^{2} - 2 \alpha \beta$