Question

Find the roots of the quadratic equation by quadratic formula :

$ab {x}^{2} + ( {b}^{2} - ac)x - bc = 0$

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Answer 1

$\huge \bf \green{Hey \: there !! }$

The given quadratic equation is abx² + ( b² - ac )x - bc = 0.

This is of the form Ax² + Bx + C = 0, where

A = ab, B = ( b² - ac ) and C = -bc .

•°• D = ( B² - 4AC )

= ( b² - ac )² + 4ab²c .

= b⁴ + a²c² - 2ab²c + 4ab²c .

= b⁴ + a²c² + 2ab²c .

= ( b² + ac )² > 0 .

Then, √D = ( b² + ac ) .


So, the given equation has real roots.

$\alpha = \frac{ -B + \sqrt{D} }{2A} \\ \\ = \frac{ - ( {b}^{2} - ac) + ( {b}^{2} + ac) }{2ab} . \\ \\ = \frac{ - \cancel{{b}^{2}} + ac + \cancel{ {b}^{2}} + ac }{2ab} \\ \\ = \frac{2ac}{2ab} = \boxed{ \purple{ \frac{c}{b} .}} \\ \\ and \\ \\ \beta = \frac{ -B - \sqrt{D} }{2A} \\ \\ = \frac{ - ( {b}^{2} - ac) - ( {b}^{2} + ac) }{2ab} \\ \\ = \frac{ - {b}^{2} + \cancel{ac} - {b}^{2} - \cancel{ac}}{2ab} \\ \\ = \frac{ - 2 {b}^{2} }{2ab} = \boxed{ \purple{ \frac{ - b}{a} .}}$



✔✔ Hence, c/b and -b/a are the roots of the given equation ✅✅.




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$\huge \bf \blue{ \#BeBrainly.}$

Answer 2

Given equation : abx^2 + ( b^2 - ac ) x - bc = 0

On comparing the given equation with ax^2 + bx + c =0, we get the following information : -

a = ab ,  b = ( b^2 - ac ) , c = - bc

Now,

       Discriminant = b^2 - 4ac

          ⇒ ( b^2 - ac )^2 - 4( ab )( - bc )

          ⇒ b^4 + ( ac )^2 - 2acb^2 + 4acb^2

          ⇒ b^4 + ( ac )^2 + 2acb^2

         

From the factorization, we know that the value of a^2 + b^2 + 2ab is ( a + b )^2

          ⇒ ( b^2 + ac )^2

        Applying quadratic equation

  $\mathsf{\implies x = \dfrac{-b\pm\sqrt{Discriminant}}{2a}}$

Substituting the values from the equation

$\implies x = \dfrac{-(b^2-ac )\pm \sqrt{(b^2+ac)^2 }}{ 2ab}\\\\\\\implies x =\dfrac{-b^2+ac\pm (b^2+ac)}{ 2ab}\\\\\\\implies x = \dfrac{ -b^2+ac+b^2+ac}{2ab}\quad\quad Or \quad \quad \dfrac{-b^2+ac-b^2-ac}{2ab}\\\\\\\implies x = \dfrac{2ac}{2ab} \quad\quad Or \quad\quad \dfrac{-2b^2}{2ab}\\\\\\\implies x = \dfrac{c}{b} \quad Or \quad - \dfrac{b}{a}$

Therefore the value of x is c /  or - b / a