Question

if alpha and beta are the zero of the polynomial f(x)=x2-2x+5 then find the quadratic polynomial whose zeroes are 1/alpha+1/beta and alpha+beta​

Answer 1

Answer:

this is the answer to your question

Answer 2

$5x^2-12x+4=0$

Step-by-step explanation:

$f(x)=x^2-2x+5$

$x=\frac{2\pm\sqrt{(-2)^2-4(1)(5)}}{2}$

By using the quadratic  formula :$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Where a=Coefficient of x square

b=Coefficient of x

c=Constant term

$x=\frac{2\pm 4i}{2}=2(\frac{1\pm 2i}{2})=1\pm 2i$

Let $\alpha=1+2i,\beta=1-2i$

Suppose, $p=\frac{1}{\alpha}+\frac{1}{\beta}=\frac{1}{1+2i}+\frac{1}{1-2i}=\frac{1-2i+1+2i}{(1+2i)(1-2i)}$

$p=\frac{2}{(1)^2-(2i)^2}=\frac{2}{1+4}=\frac{2}{5}$

Using identity:$(a+b)(a-b)=a^2-b^2$

$i^2=-1$

$q=\alpha+\beta=1+2i+1-2i=2$

General equation of quadratic polynomial

$x^2-(sum\;of\;zeroes)x+product\;of\;zeroes$

Using the formula

$x^2-(\frac{2}{5}+2)x+\frac{2}{5}\times 2$

$x^2-\frac{2+10}{5}x+\frac{4}{5}$

$\frac{5x^2-12x+4}{5}=0$

$5x^2-12x+4=0$

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