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AlgebraQuestion and Answers: Page 287

Question Number 79340 Answers: 1 Comments: 3

Question Number 79306 Answers: 1 Comments: 7

(√(3−x))−(√(x+1))>(1/2)

$$ \\ $$$$\sqrt{\mathrm{3}−\mathrm{x}}−\sqrt{\mathrm{x}+\mathrm{1}}>\frac{\mathrm{1}}{\mathrm{2}} \\ $$

Question Number 79279 Answers: 1 Comments: 4

solve ∣x∣^3 −7x^2 +7∣x∣+15<0

$$\mathrm{solve}\: \\ $$$$\mid\mathrm{x}\mid^{\mathrm{3}} −\mathrm{7x}^{\mathrm{2}} +\mathrm{7}\mid\mathrm{x}\mid+\mathrm{15}<\mathrm{0} \\ $$

Question Number 79256 Answers: 1 Comments: 0

3s^2 −2ps−3cp−1=0 and 3s−2p−sp^2 −3cp^2 =0 find s and p both real in terms of c ∈R.

$$\mathrm{3}{s}^{\mathrm{2}} −\mathrm{2}{ps}−\mathrm{3}{cp}−\mathrm{1}=\mathrm{0}\:\:\:{and} \\ $$$$\mathrm{3}{s}−\mathrm{2}{p}−{sp}^{\mathrm{2}} −\mathrm{3}{cp}^{\mathrm{2}} =\mathrm{0} \\ $$$${find}\:{s}\:{and}\:{p}\:{both}\:{real}\:{in}\:{terms} \\ $$$${of}\:{c}\:\in\mathbb{R}. \\ $$

Question Number 79249 Answers: 1 Comments: 3

Question Number 79233 Answers: 1 Comments: 3

(1/(x(x+1)))+(1/((x+1)(x+2)))+ (1/((x+2)(x+3)))≤(3/4)

$$\frac{\mathrm{1}}{\mathrm{x}\left(\mathrm{x}+\mathrm{1}\right)}+\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{1}\right)\left(\mathrm{x}+\mathrm{2}\right)}+ \\ $$$$\frac{\mathrm{1}}{\left(\mathrm{x}+\mathrm{2}\right)\left(\mathrm{x}+\mathrm{3}\right)}\leqslant\frac{\mathrm{3}}{\mathrm{4}} \\ $$

Question Number 79190 Answers: 4 Comments: 13

if x^2 +y^2 =50, find the minimum and maximum of (x+y)^2 −8(x+y)+20

$${if}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{50}, \\ $$$${find}\:{the}\:{minimum}\:{and}\:{maximum}\:{of} \\ $$$$\left({x}+{y}\right)^{\mathrm{2}} −\mathrm{8}\left({x}+{y}\right)+\mathrm{20} \\ $$

Question Number 79147 Answers: 1 Comments: 0

(√(x+(1/x^2 )))+(√(x−(1/x^2 ) ))≤(2/x)

$$\sqrt{\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }}+\sqrt{\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\:}\leqslant\frac{\mathrm{2}}{\mathrm{x}} \\ $$

Question Number 79089 Answers: 0 Comments: 2

Question Number 79085 Answers: 1 Comments: 1

α and β (or more) are root of: (x^2 +x)^2 −2(x^2 +x)−k=0 and αβ<0 ⇒k∈?

$$\alpha\:\mathrm{and}\:\beta\:\left(\mathrm{or}\:\mathrm{more}\right)\:\:\mathrm{are}\:\mathrm{root}\:\mathrm{of}: \\ $$$$\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)^{\mathrm{2}} −\mathrm{2}\left(\mathrm{x}^{\mathrm{2}} +\mathrm{x}\right)−\mathrm{k}=\mathrm{0} \\ $$$$\mathrm{and}\:\:\alpha\beta<\mathrm{0}\:\:\:\:\:\:\:\:\Rightarrow\mathrm{k}\in? \\ $$

Question Number 78948 Answers: 1 Comments: 7

Prove by mathematical induction that. n^4 + 4n^2 + 11 is divisible by 16

$$\mathrm{Prove}\:\mathrm{by}\:\mathrm{mathematical}\:\mathrm{induction}\:\mathrm{that}. \\ $$$$\:\:\:\mathrm{n}^{\mathrm{4}} \:+\:\mathrm{4n}^{\mathrm{2}} \:+\:\mathrm{11}\:\:\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{16} \\ $$

Question Number 78931 Answers: 1 Comments: 8

if x+(1/x)=a find x^n +(1/x^n )=?

$${if}\:{x}+\frac{\mathrm{1}}{{x}}={a} \\ $$$${find}\:{x}^{{n}} +\frac{\mathrm{1}}{{x}^{{n}} }=? \\ $$

Question Number 78877 Answers: 1 Comments: 6

if ((sin(A))/(sin(B)))=((sin(D))/(sin(C))) and A+B=C+D then prove that A+B=180 C+D=180

$${if}\:\:\:\frac{{sin}\left({A}\right)}{{sin}\left({B}\right)}=\frac{{sin}\left({D}\right)}{{sin}\left({C}\right)} \\ $$$${and}\:{A}+{B}={C}+{D} \\ $$$$ \\ $$$${then}\:{prove}\:{that}\:\: \\ $$$${A}+{B}=\mathrm{180} \\ $$$${C}+{D}=\mathrm{180} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$

Question Number 78857 Answers: 1 Comments: 5

with a,b∈R prove that ((a+(√3)bi))^(1/3) +((a−(√3)bi))^(1/3) has always real value and find this value (or a way how to find). examples: a=15, b=((28)/9) ⇒ result=5 a=6, b=((35)/9) ⇒ result=4 a=−24, b=((80)/9) ⇒ result=4

$${with}\:{a},{b}\in{R}\:{prove}\:{that} \\ $$$$\sqrt[{\mathrm{3}}]{{a}+\sqrt{\mathrm{3}}{bi}}+\sqrt[{\mathrm{3}}]{{a}−\sqrt{\mathrm{3}}{bi}} \\ $$$${has}\:{always}\:{real}\:{value}\:{and}\:{find}\:{this} \\ $$$${value}\:\left({or}\:{a}\:{way}\:{how}\:{to}\:{find}\right). \\ $$$${examples}: \\ $$$${a}=\mathrm{15},\:{b}=\frac{\mathrm{28}}{\mathrm{9}}\:\:\Rightarrow\:{result}=\mathrm{5} \\ $$$${a}=\mathrm{6},\:{b}=\frac{\mathrm{35}}{\mathrm{9}}\:\:\Rightarrow\:{result}=\mathrm{4} \\ $$$${a}=−\mathrm{24},\:{b}=\frac{\mathrm{80}}{\mathrm{9}}\:\:\Rightarrow\:{result}=\mathrm{4} \\ $$

Question Number 78814 Answers: 1 Comments: 1

Question Number 78820 Answers: 1 Comments: 0

please what is the fomula to determinate the equations of bissectors in triangle?

$$\mathrm{please}\:\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{fomula}\:\mathrm{to}\: \\ $$$$\mathrm{determinate}\:\mathrm{the}\:\mathrm{equations}\:\mathrm{of}\: \\ $$$$\mathrm{bissectors}\:\mathrm{in}\:\mathrm{triangle}? \\ $$

Question Number 78762 Answers: 1 Comments: 1

3acr^2 (1−r)+3apr(1−r)(pa+qb) +3bqr(1−r)(pa+qb) = 3(1−r)^2 (pa+qb)^2 +r^2 b^2 Find p, q, r such that the equation is satisfied for general any values of a,b,c.

$$\mathrm{3}{acr}^{\mathrm{2}} \left(\mathrm{1}−{r}\right)+\mathrm{3}{apr}\left(\mathrm{1}−{r}\right)\left({pa}+{qb}\right) \\ $$$$+\mathrm{3}{bqr}\left(\mathrm{1}−{r}\right)\left({pa}+{qb}\right) \\ $$$$\:\:\:=\:\mathrm{3}\left(\mathrm{1}−{r}\right)^{\mathrm{2}} \left({pa}+{qb}\right)^{\mathrm{2}} +{r}^{\mathrm{2}} {b}^{\mathrm{2}} \\ $$$${Find}\:{p},\:{q},\:{r}\:{such}\:{that}\:{the}\:{equation} \\ $$$${is}\:{satisfied}\:{for}\:{general}\:{any} \\ $$$${values}\:{of}\:{a},{b},{c}.\: \\ $$

Question Number 78755 Answers: 0 Comments: 2

Solve the equation: xy + 5x + 5y = − 25 ... (i) yz + 3y + 5z = − 15 ... (ii) xz + 5z + 3x = − 15 ... (iii)

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}: \\ $$$$\:\:\:\:\mathrm{xy}\:+\:\mathrm{5x}\:+\:\mathrm{5y}\:\:=\:\:−\:\mathrm{25}\:\:\:\:\:\:...\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\mathrm{yz}\:+\:\mathrm{3y}\:+\:\mathrm{5z}\:\:=\:\:−\:\mathrm{15}\:\:\:\:\:\:...\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\:\mathrm{xz}\:+\:\mathrm{5z}\:+\:\mathrm{3x}\:\:=\:\:−\:\mathrm{15}\:\:\:\:\:\:...\:\left(\mathrm{iii}\right) \\ $$

Question Number 78732 Answers: 0 Comments: 2

Question Number 78694 Answers: 6 Comments: 2

Solve the equation. x^2 − (y − z)^2 = 10 ... (i) y^2 − (z − x)^2 = 5 ... (ii) z^2 − (x − y)^2 = 2 ... (iii)

$$\mathrm{Solve}\:\mathrm{the}\:\mathrm{equation}. \\ $$$$\:\:\:\:\:\mathrm{x}^{\mathrm{2}} \:−\:\left(\mathrm{y}\:−\:\mathrm{z}\right)^{\mathrm{2}} \:\:=\:\:\mathrm{10}\:\:\:\:\:\:...\:\left(\mathrm{i}\right) \\ $$$$\:\:\:\:\:\mathrm{y}^{\mathrm{2}} \:−\:\left(\mathrm{z}\:−\:\mathrm{x}\right)^{\mathrm{2}} \:\:=\:\:\mathrm{5}\:\:\:\:\:\:...\:\left(\mathrm{ii}\right) \\ $$$$\:\:\:\:\:\mathrm{z}^{\mathrm{2}} \:−\:\left(\mathrm{x}\:\:−\:\mathrm{y}\right)^{\mathrm{2}} \:\:=\:\:\mathrm{2}\:\:\:\:\:\:...\:\left(\mathrm{iii}\right) \\ $$

Question Number 78650 Answers: 2 Comments: 4

Question Number 78568 Answers: 1 Comments: 0

Find the roots of the equation bx^3 − (3b + 2)x^2 − 2(5b − 3)x + 20 = 0

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\mathrm{bx}^{\mathrm{3}} \:−\:\left(\mathrm{3b}\:+\:\mathrm{2}\right)\mathrm{x}^{\mathrm{2}} \:−\:\mathrm{2}\left(\mathrm{5b}\:−\:\mathrm{3}\right)\mathrm{x}\:+\:\mathrm{20}\:\:=\:\:\mathrm{0} \\ $$

Question Number 78563 Answers: 0 Comments: 0

Question Number 78503 Answers: 3 Comments: 0

if x,y >1 prove (x^2 /(y−1))+(y^2 /(x−1))≥8

$${if}\:{x},{y}\:>\mathrm{1}\: \\ $$$${prove}\:\frac{{x}^{\mathrm{2}} }{{y}−\mathrm{1}}+\frac{{y}^{\mathrm{2}} }{{x}−\mathrm{1}}\geqslant\mathrm{8} \\ $$

Question Number 78465 Answers: 1 Comments: 0

Question Number 78460 Answers: 3 Comments: 3

Σ_(n=1) ^∞ (n^3 /3^n )=?

$$\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\frac{{n}^{\mathrm{3}} }{\mathrm{3}^{{n}} }=? \\ $$

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