Question

if alpha and beta are the zeroes of the quadratic polynomial f(x) = x^2-5x+4 find a^2+b^2

Answer 1

p(x)= x^2 - 5x +4

Sum of zeroes = -b/a

(a + b) = - (-5)

=5

Product of zeroes = c/a

ab =4

=a^2 + b^2

= (a + b) ^2 -2ab

= (5)^2 - 2(4)

= 25 - 8

=17

Answer 2

$since \: \alpha \: and \: \beta \: are \: the \: zeros \: of \: the \: given \: p(x)$

Sum of zeros = -b/a

$\alpha + \beta = - \frac{ - 5}{1}$

$\alpha + \beta = 5$

Product of zeros = c/a

$\alpha \beta = \frac{4}{1}$

$\alpha \beta = 4$

Now,

$using \: (a + b)^{2} = {a}^{2} + {b}^{2} + 2ab$

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

${5}^{2} = { \alpha }^{2} + { \beta }^{2} + 2 \times 4$

$25 = { \alpha }^{2} + { \beta }^{2} + 8$

${ \alpha }^{2} + { \beta }^{2} = 25 - 8$

${ \alpha }^{2} + { \beta }^{2} = 17$