[tex] \huge \boxed{\mathfrak{Question} \downarrow}[/tex]
Solve the equation using the quadratic formula ⇨ x² + 11x + 9 = 0
[tex] \large \boxed{\mathfrak{Answer \: with \: Explanation} \downarrow}[/tex]
[tex] \sf \: x ^ { 2 } + 11 x + 9 = 0[/tex]
All equations of the form [tex]\sf\:ax^{2}+bx+c=0[/tex] can be solved using the quadratic formula: [tex]\sf\frac{-b±\sqrt{b^{2}-4ac}}{2a}[/tex]. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.
[tex] \sf \: x^{2}+11x+9=0 [/tex]
This equation is in standard form: ax² + bx + c = 0. Substitute 1 for a, 11 for b and 9 for c in the quadratic formula [tex]\sf\frac{-b±\sqrt{b^{2}-4ac}}{2a}[/tex].
[tex] \sf \: x=\frac{-11±\sqrt{11^{2}-4\times 9}}{2} \\ [/tex]
Square 11.
[tex] \sf \: x=\frac{-11±\sqrt{121-4\times 9}}{2} \\ [/tex]
Multiply -4 times 9.
[tex] \sf \: x=\frac{-11±\sqrt{121-36}}{2} \\ [/tex]
Add 121 to -36.
[tex] \sf \: x=\frac{-11±\sqrt{85}}{2} \\ [/tex]
Now solve the equation [tex]\sf\:x=\frac{-11±\sqrt{85}}{2}[/tex] when ± is plus. Add -11 to √85.
[tex] \boxed{ \boxed{\bf \: x=\frac{\sqrt{85}-11}{2} }}[/tex]
Now solve the equation [tex]\sf\:x=\frac{-11±\sqrt{85}}{2}[/tex] when ± is minus. Subtract √85 from -11.
[tex] \boxed{ \boxed{\bf \: x=\frac{-\sqrt{85}-11}{2}} } \\ [/tex]
The equation is now solved. The solution set is :-
[tex]\bf \: x=\frac{\sqrt{85}-11}{2} \\ \\ \sf \: and \\ \\ \bf \: x=\frac{-\sqrt{85}-11}{2} [/tex]
