Question

If a and b are zeroes and the quadratic polynomial f(x) = x² -x -4, then the value of 1/a + 1/b = ab is​

Answer 1

SOLUTION

GIVEN

a and b are zeroes and the quadratic polynomial

f(x) = x² - x - 4

TO DETERMINE

The value of

$\displaystyle \sf{ \frac{1}{a} + \frac{1}{b} - ab}$

FORMULA TO BE IMPLEMENTED

If $\alpha \: \: and \: \: \beta \:$are the zeroes of the quadratic polynomial $a {x}^{2} + bx + c$

Then

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

EVALUATION

Here it is given that a and b are zeroes and the quadratic polynomial

f(x) = x² - x - 4

Sum of the zeroes = a + b = 1

Product of the zeroes = ab = - 4

Now

$\displaystyle \sf{ \frac{1}{a} + \frac{1}{b} - ab}$

$\displaystyle \sf{ = \frac{b + a}{ab} - ab}$

$\displaystyle \sf{ = \frac{a + b}{ab} - ab}$

$\displaystyle \sf{ = \frac{1}{ - 4} - ( - 4)}$

$\displaystyle \sf{ = - \frac{1}{ 4} + 4}$

$\displaystyle \sf{ =4 - \frac{1}{ 4} }$

$\displaystyle \sf{ = \frac{16 - 1}{ 4} }$

$\displaystyle \sf{ = \frac{15}{ 4} }$

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