[tex] \star \; {\underline{\underline{\pmb{\purple{\frak{ \; Question \; :- }}}}}} [/tex]
[tex] \longmapsto [/tex] Solve the following using Quadratic formula and find the Roots :
i) 4x² + 4bx - (a² - b²) = 0
ii) √2x² + 7x + 5√2 = 0
iii) [tex] \sf { \dfrac{x + 3}{x - 2} - \dfrac{1 - x}{x} = \dfrac{17}{4} } [/tex]
[tex] \\ \\ [/tex]
[tex] \longrightarrow \red{\sf{ \; No \; Spamming }} [/tex]
[tex] \longrightarrow \green{\sf{ \; Answer \; With \; Proper \; Explaination }} [/tex]
[tex] \longrightarrow \pink{\sf{ \; Thank \; You \; !! }} [/tex]
[tex] \\ {\underline{\rule{300pt}{9pt}}} [/tex]
[tex] \longmapsto [/tex] Solve the following using Quadratic formula and find the Roots :
i) 4x² + 4bx - (a² - b²) = 0
ii) √2x² + 7x + 5√2 = 0
iii) [tex] \sf { \dfrac{x + 3}{x - 2} - \dfrac{1 - x}{x} = \dfrac{17}{4} } [/tex]
[tex] \\ \\ [/tex]
[tex] \longrightarrow \red{\sf{ \; No \; Spamming }} [/tex]
[tex] \longrightarrow \green{\sf{ \; Answer \; With \; Proper \; Explaination }} [/tex]
[tex] \longrightarrow \pink{\sf{ \; Thank \; You \; !! }} [/tex]
[tex] \\ {\underline{\rule{300pt}{9pt}}} [/tex]
Answers & Comments
Verified answer
[tex]\large \underline{\boxed{\sf{ Given - }}}[/tex]
[tex]\large \underline{\boxed{\sf{To \: Find- }}}[/tex]
[tex]\large \underline{\boxed{\sf{Solution- }}}[/tex]
Given equation is-
[tex]\large{\sf{ \implies {4x}^{2} + 4bx - ( {a}^{2} - {b}^{2} ) = 0 }}[/tex]
[tex]\large{\sf{ \implies {4x}^{2} + 4bx - {a}^{2} + {b}^{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies {4x}^{2} + 4bx + {b}^{2} - {a}^{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies( {4x}^{2} + 4bx + {b}^{2} ) - {a}^{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies \bigg \{{(2x)}^{2} + 2 \times 2 \times b \times x + {b}^{2} \bigg \} - {a}^{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies {(2x + b)}^{2} - {a}^{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies \bigg \{(2x + b + a) (2x + b - a) \bigg \}= 0 }}[/tex]
[tex]\large{\sf{ \implies(2x + b + a) = 0 \: ; \: (2x + b - a)= 0 }}[/tex]
[tex]\large{\sf{ \implies(2x + b) = - a \: ; \: (2x + b) = a}}[/tex]
[tex]\large{\sf{ \implies2x = ( - a - b) \: ; \: 2x = (a - b)}}[/tex]
[tex]\large \displaystyle{\sf{ \implies x = \frac{ ( - a - b)}{2} \: ; \: x = \frac{(a - b)}{2}}}[/tex]
[tex]\large \displaystyle{\sf{ \implies x = - \frac{ ( a + b)}{2} \: ; \: x = \frac{(a - b)}{2}}}[/tex]
[tex] \rule100mm3pt[/tex]
[tex]\large \underline{\boxed{\sf{ Given - }}}[/tex]
[tex]\large \underline{\boxed{\sf{To \: Find- }}}[/tex]
[tex]\large \underline{\boxed{\sf{Solution- }}}[/tex]
[tex]\large{\sf{ \implies \sqrt{2} {x}^{2} + 7x + 5 \sqrt{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies \sqrt{2} {x}^{2} + (5 + 2)x+ 5 \sqrt{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies \sqrt{2} {x}^{2} + 5x + 2x+ 5 \sqrt{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies \sqrt{2} {x}^{2} + 2x + 5x+ 5 \sqrt{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies \sqrt{2} {x}^{2} + ( \sqrt{2})^{2} x + 5x+ 5 \sqrt{2} = 0 }}[/tex]
[tex]\large{\sf{ \implies x\sqrt{2} (x + \sqrt{2} ) + 5(x + \sqrt{2} )= 0 }}[/tex]
[tex]\large{\sf{ \implies (x \sqrt{2} + 5) (x + \sqrt{2} )= 0 }}[/tex]
[tex]\large{\sf{ \implies (x \sqrt{2} + 5) \: \: ; \: \: (x + \sqrt{2} )= 0 }}[/tex]
[tex]\large{\sf{ \implies (x \sqrt{2} + 5) = 0 \: \: ; \: \: (x + \sqrt{2} )= 0 }}[/tex]
[tex]\large{\sf{ \implies x \sqrt{2} = - 5 \: \: ; \: \: x = - \sqrt{2} }}[/tex]
[tex]\large \displaystyle{\sf{ \implies x = \frac{ - 5 }{ \sqrt{2} } \: \: ; \: \: x = - \sqrt{2} }}[/tex]
[tex] \rule100mm3pt[/tex]
[tex]\large \underline{\boxed{\sf{ Given - }}}[/tex]
[tex]\large \underline{\boxed{\sf{Solution- }}}[/tex]
[tex]\large \displaystyle{\sf{ \implies \frac{x + 3}{x - 2} - \frac{1 - x}{x} = \frac{17}{4} }}[/tex]
[tex]\large \displaystyle{\sf{ \implies \frac{x(x + 3)}{x(x - 2)} - \frac{(x - 2)(1 - x)}{x(x - 2)} = \frac{17}{4} }}[/tex]
[tex]\large \displaystyle{\sf{ \implies \frac{ {x}^{2} + 3x - (x - 2)(1 - x)}{x(x - 2)} = \frac{17}{4} }}[/tex]
[tex]\large \displaystyle{\sf{ \implies \frac{ {x}^{2} + 3x - x(1 - x) - 2(1 - x)}{x(x - 2)} = \frac{17}{4} }}[/tex]
[tex]\large \displaystyle{\sf{ \implies \frac{ {x}^{2} + 3x - x + {x}^{2} - 2 - 2x}{x(x - 2)} = \frac{17}{4} }}[/tex]
[tex]\large \displaystyle{\sf{ \implies \frac{ {x}^{2} + 3x - 3x + {x}^{2} - 2 }{ {x}^{2} - 2x} = \frac{17}{4} }}[/tex]
[tex]\large \displaystyle{\sf{ \implies \frac{{2x}^{2} - 2 }{ {x}^{2} - 2x} = \frac{17}{4} }}[/tex]
[tex]\large \displaystyle{\sf{ \implies 4({2x}^{2} - 2 )}{= {17( {x}^{2} - 2x) } }}[/tex]
[tex]\large \displaystyle{\sf{ \implies ({8x}^{2} - 8 )}{= ( {17x}^{2} - 34x) }}[/tex]
[tex]\large \displaystyle{\sf{ \implies ( - {8x}^{2} - 8 + {17x}^{2} + 34 ) = 0}}[/tex]
[tex]\large \displaystyle{\sf{ \implies ( - 8 + {9x}^{2} + 34x) = 0}}[/tex]
[tex]\large \displaystyle{\sf{ \implies ( {9x}^{2} + 34x - 8) = 0}}[/tex]
[tex]\large \displaystyle{\sf{ \implies ( {9x}^{2} + 36x - 2x - 8) = 0}}[/tex]
[tex]\large \displaystyle{\sf{ \implies \bigg \{ {9x}(x + 4) - 2(x + 4) \bigg \}= 0}}[/tex]
[tex]\large \displaystyle{\sf{ \implies (9x - 2) =0\: \: ; \: \: (x + 4)= 0}}[/tex]
[tex]\large \displaystyle{\sf{ \implies 9x = 2\: \: ; \: \: x = - 4}}[/tex]
[tex]\large \displaystyle{\sf{ \implies x = \frac{2}{9}\: \: ; \: \: x = - 4}}[/tex]
[tex] \rule100mm3pt[/tex]
Explanation :-
[tex]:: \implies{\mathtt{{4 {x}^{2} + 4bx - ({a}^{2} - {b}^{2}) = 0 }}}[/tex]
[tex]:: \implies{\mathtt{{4 {x}^{2} + 4bx + {b}^{2} - {a}^{2} = 0 }}}[/tex]
[tex]:: \implies{\mathtt{{(2x + {b}^{2})^{2} - {a}^{2} = 0 }}}[/tex]
[tex]:: \implies{\mathtt{{(2x+b−a)(2x+b+a)=0 }}}[/tex]
[tex]:: \implies{\mathtt{{2x+b−a=0 }}}[/tex]
[tex]:: \implies{\mathtt{{Or, 2x+b+a=0}}}[/tex]
[tex] \star{\large{ \sf{ \underline{ \underline{ \purple { \boxed{ \pmb{ \frak{{x = \frac{a+b}{2} ; \frac{a - b}{2} }}}}}}}}}}[/tex]
[tex] \begin{gathered} \\ \qquad{ \rule{250pt}{2.5pt}} \\ \end{gathered}[/tex]
[tex]:: \implies{\mathtt{{ \sqrt{2} {x}^{2} + 5x + 2x + 5 \sqrt{2} = 0 }}}[/tex]
[tex] \\ :: \implies{ \mathtt{\sqrt{2} {x}^{2} + 5x + \sqrt{2} \times \sqrt{2} \times x + 5\sqrt{2} = 0 }}[/tex]
[tex] \\ :: \implies{ \mathtt{x( \sqrt{2}x + 5 ) + \sqrt{2}(\sqrt{2}x + 5) = 0}}[/tex]
[tex] \\ :: \implies{ \mathtt{( \sqrt{2} + 5)(x + \sqrt{2}) = 0 }}[/tex]
[tex] \\ :: \implies{ \mathtt{x = \frac{5}{ \sqrt{2} } }}[/tex]
[tex] \star{ \large{ \green{ \underline{ \underline{ \boxed{ \pmb{ \frak{x = - \sqrt{2} }}}}}}}}[/tex]
[tex] \begin{gathered} \\ \qquad{ \rule{250pt}{2.5pt}} \\ \end{gathered}[/tex]
[tex] \\ :: \implies{\mathtt{{ x(x + 3) - (1 - x) (\frac{x - 2}{x})(x - 2) = \frac{17}{4} }}}[/tex]
[tex] \\ :: \implies{\mathtt{{ 2 {x}^{2} + \frac{2}{ {x}^{2} } - 2x = \frac{17}{4} }}}[/tex]
[tex] \\ :: \implies{\mathtt{{ 8 {x}^{2} + 8 = 17 {x}^{2} - 34x }}}[/tex]
[tex] \\ :: \implies{\mathtt{ \bold{ 9 {x}^{2} - 34x - 8 = 0 }}}[/tex]
Now,
We will use the method of factorisation
[tex] \\ :: \implies{\mathtt{{ 9x^2-36x+2x-8=0 }}}[/tex]
[tex] \\ :: \implies{\mathtt{{9x(x-4)+2(x-4) =0 }}}[/tex]
[tex] \\ :: \implies{\mathtt{ {(x-4)(9x+2)=0 }}}[/tex]
[tex] \star{ \large{ \blue{ \underline{ \underline{ \boxed{ \pmb{ \frak{x = -4=0 }}}}}}}}[/tex]
x = 4,[tex]\frac{-2}{9}[/tex]
[tex] \begin{gathered} \\ \qquad{ \rule{250pt}{2.5pt}} \\ \end{gathered}[/tex]