Question

solve the quadratic equation by using formula method 2x²-9x-5=0​

Answer 1

EXPLANATION.

Quadratic equation.

⇒ 2x² - 9x - 5 = 0.

As we know that,

Method = 1.

Factorizes the equation into middle term splits, we get.

⇒ 2x² - 10x + x - 5 = 0.

⇒ 2x(x - 5) + 1(x - 5) = 0.

⇒ (2x + 1)(x - 5) = 0.

⇒ x = -1/2  and  x = 5.

Method = 2.

As we know that,

⇒ D = Discriminant or b² - 4ac.

⇒ D = (-9)² - 4(2)(-5).

⇒ D = 81 + 40.

⇒ D = 121.

⇒ x = -b + √D/2a.

⇒ y = -b - √D/2a.

⇒ x = -(-9) + √121/2(2).

⇒ x = 9 + 11/4.

⇒ x = 20/4.

⇒ x = 5.

⇒ y = -(-9) - √121/2(2).

⇒ y = 9 - 11/4.

⇒ y = -2/4.

⇒ y = -1/2.

                                                                                                                     

MORE INFORMATION.

Conjugate roots.

If D < 0.

One roots = α + iβ.

Other roots = α - iβ.

If D > 0.

One roots = α + √β.

Other roots = α - √β.

Answer 2

${ \red{ \tt{ \underline{ \underline{ \huge{QUESTION}}}}}}$

${ \sf{ \: solve \: the \: quadratic \: equation }} \\ { \sf{\: by \: using \: the \: formula \: method.... }} \\ { \boxed { \boxed{ \green{ \tt{2 {x}^{2} - 9x - 5 = 0}}}}}$

${ \red{ \underline{ \underline{ \tt{ \huge{SOLUTION}}}}}}$

${ \sf{ \blue{ using \: the \: quadratic \: formula \: also \: known \: as \: }}} \\ {\underline{\green{\sf{Sridhar \: Acharya 's \: formula}}}} \\ \\ { \dashrightarrow{ \boxed{ \boxed{ \red{ \tt{x = \frac{ - b± \: \sqrt{ {b}^{2} - 4ac } }{2a} }}}}}}$

${ \sf{in \: the \: given \: question}} \\ { \dashrightarrow{ \tt{ \green{a = 2}}}} \\ { \dashrightarrow{ \tt{ \green{b = - 9}}}} \\ { \dashrightarrow{ \tt{ \green{c = - 5}}}} \\ \\ { \sf{ \blue{ subsituting \: the \: above \: values \: in \: the \: formula}}}$

${ : { \implies{ \tt{ \green{x = \frac{ - ( - 9)± \sqrt{ {9}^{2} - 4(2)( - 5) } }{2(2)}}}}}}$

${ :{ \implies{ \tt{ \green{x = \frac{9± \sqrt{(81 + 40)} }{4}}}}}} \\ \\ \\ { : { \implies{ \tt{ \green{ x = \frac{9± \sqrt{121} }{4} }}}}} \\ \\ \\ { : { \implies{ \tt{ \green{x = \frac{9±11}{4}}}}}}$

${ \underline{ \red{ \sf{first \: value \: of \: root}}}} \\ \\ { :{ \implies{ \tt{ \green{x = \frac{9 + 11}{4} }}}}} \\ \\ { :{ \implies{ \tt{ \green{x = \frac{20}{4}}}}}} \\ \\ { :{ \implies{ \underline{ \boxed{ \tt{ \green{x = 5}}}}}}}$

${ \underline{ \sf{ \red{second \: value \: of \: root}}}} \\ \\ { :{ \implies{ \tt{ \green{x = \frac{9 - 11}{4}}}}}} \\ \\ { :{ \implies{ \tt{ \green{x = \frac{ - 2}{4}}}}}} \\ \\ { :{ \implies{ \underline{ \boxed{ \tt{ \green{x = \frac{ - 1}{2 }}}}}}}}$