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Developed the formula for the cubic equation when it has three real roots and Cardano is just not suitable as discriminant gets negative.. Given x^3 −bx−c=0 with b>0 , c>0 (t/b)=1+((15)/2)((c^2 /b^3 ))+(√(((c^2 /b^3 ))[49−((75)/4)((c^2 /b^3 ))])) x=(√(t+((4c^2 )/b^2 ))) −((3c)/b) ★ mrW Sir, MjS Sir please check!

$${Developed}\:{the}\:{formula}\:{for}\:{the}\:{cubic} \\ $$$${equation}\:{when}\:{it}\:{has}\:{three}\:{real}\:{roots} \\ $$$${and}\:{Cardano}\:{is}\:{just}\:{not}\:{suitable}\:{as} \\ $$$${discriminant}\:{gets}\:{negative}.. \\ $$$$\:\:\:{Given}\:\:\:\:\boldsymbol{{x}}^{\mathrm{3}} −\boldsymbol{{bx}}−\boldsymbol{{c}}=\mathrm{0} \\ $$$$\:\:{with}\:\:\boldsymbol{{b}}>\mathrm{0}\:,\:\boldsymbol{{c}}>\mathrm{0} \\ $$$$\:\frac{\boldsymbol{{t}}}{\boldsymbol{{b}}}=\mathrm{1}+\frac{\mathrm{15}}{\mathrm{2}}\left(\frac{\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{b}}^{\mathrm{3}} }\right)+\sqrt{\left(\frac{\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{b}}^{\mathrm{3}} }\right)\left[\mathrm{49}−\frac{\mathrm{75}}{\mathrm{4}}\left(\frac{\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{b}}^{\mathrm{3}} }\right)\right]} \\ $$$$\:\:\:\:\:\boldsymbol{{x}}=\sqrt{\boldsymbol{{t}}+\frac{\mathrm{4}\boldsymbol{{c}}^{\mathrm{2}} }{\boldsymbol{{b}}^{\mathrm{2}} }}\:−\frac{\mathrm{3}\boldsymbol{{c}}}{\boldsymbol{{b}}}\:\:\bigstar \\ $$$${mrW}\:{Sir},\:{MjS}\:{Sir}\:\:{please}\:{check}! \\ $$

Question Number 114189 Answers: 0 Comments: 6

Question Number 114134 Answers: 1 Comments: 4

(m^2 −n^2 +6(n+m)/(m^2 −(6−n)^2 m+n=12

$$\left({m}^{\mathrm{2}} −{n}^{\mathrm{2}} +\mathrm{6}\left({n}+{m}\right)/\left({m}^{\mathrm{2}} −\left(\mathrm{6}−{n}\right)^{\mathrm{2}} \right.\right. \\ $$$${m}+{n}=\mathrm{12} \\ $$

Question Number 114110 Answers: 1 Comments: 2

(4/(1!)) + ((11)/(2!)) + ((22)/(3!)) + ((37)/(4!)) + ... = ?

$$\frac{\mathrm{4}}{\mathrm{1}!}\:+\:\frac{\mathrm{11}}{\mathrm{2}!}\:+\:\frac{\mathrm{22}}{\mathrm{3}!}\:+\:\frac{\mathrm{37}}{\mathrm{4}!}\:+\:...\:=\:? \\ $$

Question Number 114077 Answers: 1 Comments: 0

Question Number 114023 Answers: 2 Comments: 1

Σ_(n=1) ^∝ (1/(n^2 −1))=?

$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\propto} {\sum}}\frac{\mathrm{1}}{\mathrm{n}^{\mathrm{2}} −\mathrm{1}}=? \\ $$

Question Number 113997 Answers: 2 Comments: 0

Question Number 113996 Answers: 1 Comments: 1

Question Number 113975 Answers: 1 Comments: 0

Find the nth term: 1, 0, − 1, 0, 1, 0, − 1, 0, ...

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{nth}\:\mathrm{term}:\:\:\:\mathrm{1},\:\:\mathrm{0},\:\:−\:\mathrm{1},\:\:\mathrm{0},\:\:\mathrm{1},\:\:\mathrm{0},\:\:−\:\mathrm{1},\:\:\mathrm{0},\:\:... \\ $$

Question Number 113931 Answers: 1 Comments: 2

∣ x ∣=2⇒ x=2 ∨ x=−2 OR ∣ x ∣=2⇒ x=2 ∧ x=−2 ?

$$\mid\:{x}\:\mid=\mathrm{2}\Rightarrow\:{x}=\mathrm{2}\:\vee\:{x}=−\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{OR} \\ $$$$\mid\:{x}\:\mid=\mathrm{2}\Rightarrow\:{x}=\mathrm{2}\:\wedge\:{x}=−\mathrm{2} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:? \\ $$

Question Number 113913 Answers: 3 Comments: 0

Find the square root of (√(50))+(√(48))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{square}\:\mathrm{root}\:\mathrm{of}\:\sqrt{\mathrm{50}}+\sqrt{\mathrm{48}} \\ $$

Question Number 113894 Answers: 2 Comments: 1

Question Number 113885 Answers: 1 Comments: 0

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Question Number 113876 Answers: 4 Comments: 0

If 2^x =4^y =8^z and xyz=288, then find (1/(2x))+(1/(4y))+(1/(8z))

$$\mathrm{If}\:\mathrm{2}^{\mathrm{x}} =\mathrm{4}^{\mathrm{y}} =\mathrm{8}^{\mathrm{z}} \:\mathrm{and}\:\mathrm{xyz}=\mathrm{288},\:\mathrm{then}\:\mathrm{find} \\ $$$$\frac{\mathrm{1}}{\mathrm{2x}}+\frac{\mathrm{1}}{\mathrm{4y}}+\frac{\mathrm{1}}{\mathrm{8z}} \\ $$

Question Number 113854 Answers: 1 Comments: 0

if f(x)=2x^2 −12x+10. (i) sketch the graph of y=∣f(x)∣ for −1≤x≤7. (ii) find the set of values of k for which the equation ∣f(x)∣=k has 4 distinct roots.

$${if}\:{f}\left({x}\right)=\mathrm{2}{x}^{\mathrm{2}} −\mathrm{12}{x}+\mathrm{10}.\: \\ $$$$\left({i}\right)\:{sketch}\:{the}\:{graph}\:{of}\:{y}=\mid{f}\left({x}\right)\mid\:{for} \\ $$$$−\mathrm{1}\leqslant{x}\leqslant\mathrm{7}. \\ $$$$\left({ii}\right)\:{find}\:{the}\:{set}\:{of}\:{values}\:{of}\:{k}\:{for} \\ $$$${which}\:{the}\:{equation}\:\mid{f}\left({x}\right)\mid={k}\:{has}\:\mathrm{4} \\ $$$${distinct}\:{roots}. \\ $$

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