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AlgebraQuestion and Answers: Page 229

Question Number 130257 Answers: 0 Comments: 2

n is an integer prove algebraically that the sum of (1/2)n(n+1) and (1/2)(n+1)(n+2) is always a square number. note: write your expression in a form that clearly shows a square number.

$${n}\:\mathrm{is}\:\mathrm{an}\:\mathrm{integer} \\ $$$$\mathrm{prove}\:\mathrm{algebraically}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}{n}\left({n}+\mathrm{1}\right)\:\mathrm{and}\:\frac{\mathrm{1}}{\mathrm{2}}\left({n}+\mathrm{1}\right)\left({n}+\mathrm{2}\right)\:\mathrm{is}\:\mathrm{always} \\ $$$$\mathrm{a}\:\mathrm{square}\:\mathrm{number}. \\ $$$$\mathrm{note}:\:{write}\:{your}\:{expression}\:{in}\:{a}\:{form} \\ $$$${that}\:{clearly}\:{shows}\:{a}\:{square}\:{number}. \\ $$

Question Number 130242 Answers: 0 Comments: 0

Find m n=(p_1 p_2 )^m 2n=(p_1 ^(1+log_(p_1 p_2 ) n) )(p_2 ^(1+log_(p_1 p_2 ) n) )

$$\mathrm{Find}\:\:\:{m} \\ $$$${n}=\left({p}_{\mathrm{1}} {p}_{\mathrm{2}} \right)^{{m}} \\ $$$$\mathrm{2}{n}=\left({p}_{\mathrm{1}} ^{\mathrm{1}+{log}_{{p}_{\mathrm{1}} {p}_{\mathrm{2}} } {n}} \right)\left({p}_{\mathrm{2}} ^{\mathrm{1}+{log}_{{p}_{\mathrm{1}} {p}_{\mathrm{2}} } {n}} \right) \\ $$

Question Number 130092 Answers: 1 Comments: 1

Question Number 130089 Answers: 1 Comments: 2

To solve x^3 =x+c Let y=(x−p)(x^3 −x−c) (dy/dx)=(x^3 −x−c)+(3x^2 −1)(x−p) let (dy/dx)=mx ⇒ (3x^2 −1)(x−p)=mx 3x^3 −3px^2 −(m+1)x+p=0 3(x+c)−3px^2 −(m+1)x+p=0 3px^2 +(m−2)x−(3c+p)=0 and since x^3 =x+c (m−2)x^2 +(2p−3c)x+3cp=0 [9cp^2 +(3c+p)(m−2)]x +[3cp(m−2)+(3c+p)(2p−3c)] =0 x=−(((3c+p)(2p−3c)+3cp(m−2))/(9cp^2 +(3c+p)(m−2))) (m−2)x^2 +(2p−3c)x+(3cp)=0 Now choosing suitable ′p′ value we find ′m′; and then x=f(p,m) ★

$${To}\:{solve}\:\:\:{x}^{\mathrm{3}} ={x}+{c} \\ $$$${Let}\:\:{y}=\left({x}−{p}\right)\left({x}^{\mathrm{3}} −{x}−{c}\right) \\ $$$$\:\:\frac{{dy}}{{dx}}=\left({x}^{\mathrm{3}} −{x}−{c}\right)+\left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}−{p}\right) \\ $$$$\:\:{let}\:\:\:\frac{{dy}}{{dx}}={mx} \\ $$$$\Rightarrow\:\:\left(\mathrm{3}{x}^{\mathrm{2}} −\mathrm{1}\right)\left({x}−{p}\right)={mx} \\ $$$$\mathrm{3}{x}^{\mathrm{3}} −\mathrm{3}{px}^{\mathrm{2}} −\left({m}+\mathrm{1}\right){x}+{p}=\mathrm{0} \\ $$$$\mathrm{3}\left({x}+{c}\right)−\mathrm{3}{px}^{\mathrm{2}} −\left({m}+\mathrm{1}\right){x}+{p}=\mathrm{0} \\ $$$$\mathrm{3}{px}^{\mathrm{2}} +\left({m}−\mathrm{2}\right){x}−\left(\mathrm{3}{c}+{p}\right)=\mathrm{0} \\ $$$${and}\:{since}\:{x}^{\mathrm{3}} ={x}+{c} \\ $$$$\left({m}−\mathrm{2}\right){x}^{\mathrm{2}} +\left(\mathrm{2}{p}−\mathrm{3}{c}\right){x}+\mathrm{3}{cp}=\mathrm{0} \\ $$$$\left[\mathrm{9}{cp}^{\mathrm{2}} +\left(\mathrm{3}{c}+{p}\right)\left({m}−\mathrm{2}\right)\right]{x} \\ $$$$+\left[\mathrm{3}{cp}\left({m}−\mathrm{2}\right)+\left(\mathrm{3}{c}+{p}\right)\left(\mathrm{2}{p}−\mathrm{3}{c}\right)\right] \\ $$$$=\mathrm{0} \\ $$$${x}=−\frac{\left(\mathrm{3}{c}+{p}\right)\left(\mathrm{2}{p}−\mathrm{3}{c}\right)+\mathrm{3}{cp}\left({m}−\mathrm{2}\right)}{\mathrm{9}{cp}^{\mathrm{2}} +\left(\mathrm{3}{c}+{p}\right)\left({m}−\mathrm{2}\right)} \\ $$$$\left({m}−\mathrm{2}\right){x}^{\mathrm{2}} +\left(\mathrm{2}{p}−\mathrm{3}{c}\right){x}+\left(\mathrm{3}{cp}\right)=\mathrm{0} \\ $$$${Now}\:{choosing}\:{suitable}\:'{p}'\:{value} \\ $$$${we}\:{find}\:'{m}'; \\ $$$${and}\:{then}\:{x}={f}\left({p},{m}\right)\:\bigstar \\ $$

Question Number 130036 Answers: 2 Comments: 0

If { ((a+b+c = 1)),((a^3 +b^3 +c^3 = 4)) :} find (1/(a+bc)) + (1/(b+ca)) + (1/(c+ab)).

$$\mathrm{If}\:\begin{cases}{{a}+{b}+{c}\:=\:\mathrm{1}}\\{{a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}} \:=\:\mathrm{4}}\end{cases} \\ $$$$\:\mathrm{find}\:\frac{\mathrm{1}}{{a}+{bc}}\:+\:\frac{\mathrm{1}}{{b}+{ca}}\:+\:\frac{\mathrm{1}}{{c}+{ab}}. \\ $$

Question Number 130006 Answers: 2 Comments: 0

Question Number 129979 Answers: 3 Comments: 0

If { ((∣x∣ + x + y = 5)),((x + ∣y∣ −y = 10)) :} then x+y ?

$$\mathrm{If}\:\begin{cases}{\mid\mathrm{x}\mid\:+\:\mathrm{x}\:+\:\mathrm{y}\:=\:\mathrm{5}}\\{\mathrm{x}\:+\:\mid\mathrm{y}\mid\:−\mathrm{y}\:=\:\mathrm{10}}\end{cases}\:\mathrm{then}\:\mathrm{x}+\mathrm{y}\:? \\ $$

Question Number 129841 Answers: 1 Comments: 2

{_((2^(2x) )^(1/3) +(2^(4x) )^(1/3) +4^x =5) ^(((2^x ))^(1/3) −2^x =2) }⇒2^x −8^x =? pleas solve thes

$$\left\{_{\sqrt[{\mathrm{3}}]{\mathrm{2}^{\mathrm{2x}} }+\sqrt[{\mathrm{3}}]{\mathrm{2}^{\mathrm{4x}} }+\mathrm{4}^{\mathrm{x}} =\mathrm{5}} ^{\sqrt[{\mathrm{3}}]{\mathrm{2}^{\mathrm{x}} \:}−\mathrm{2}^{\mathrm{x}} =\mathrm{2}} \right\}\Rightarrow\mathrm{2}^{\mathrm{x}} −\mathrm{8}^{\mathrm{x}} =? \\ $$$$\mathrm{pleas}\:\mathrm{solve}\:\mathrm{thes} \\ $$

Question Number 129832 Answers: 1 Comments: 2

Question Number 129687 Answers: 1 Comments: 0

x^3 −x−c=0 (solve for x even if 0<c<(2/(3(√3))))

$$\:\:\:\:{x}^{\mathrm{3}} −{x}−{c}=\mathrm{0} \\ $$$$\left({solve}\:{for}\:{x}\:\:{even}\:{if}\:\:\:\mathrm{0}<{c}<\frac{\mathrm{2}}{\mathrm{3}\sqrt{\mathrm{3}}}\right) \\ $$

Question Number 129677 Answers: 0 Comments: 2

Question Number 129663 Answers: 0 Comments: 4

(√(14+(√3) +((26−5(√3)))^(1/3) )) =?

$$\:\sqrt{\mathrm{14}+\sqrt{\mathrm{3}}\:+\sqrt[{\mathrm{3}}]{\mathrm{26}−\mathrm{5}\sqrt{\mathrm{3}}}}\:=?\: \\ $$

Question Number 129622 Answers: 1 Comments: 2

Let f(x)=(x+1)^2 for all x≥−1. If g(x) is the function whose graph is the reflection of the graph of f(x) with respect to the line y=x, then g(x) is equal to...

$$\mathrm{Let}\:\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} \:\mathrm{for}\:\mathrm{all}\:\mathrm{x}\geqslant−\mathrm{1}.\:\mathrm{If}\:\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{the} \\ $$$$\mathrm{function}\:\mathrm{whose}\:\mathrm{graph}\:\mathrm{is}\:\mathrm{the}\:\mathrm{reflection}\:\mathrm{of}\:\mathrm{the}\:\mathrm{graph} \\ $$$$\mathrm{of}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{with}\:\mathrm{respect}\:\mathrm{to}\:\mathrm{the}\:\mathrm{line}\:\mathrm{y}=\mathrm{x},\:\mathrm{then}\:\mathrm{g}\left(\mathrm{x}\right)\:\mathrm{is} \\ $$$$\mathrm{equal}\:\mathrm{to}... \\ $$

Question Number 129607 Answers: 2 Comments: 0

salom

$${salom} \\ $$

Question Number 129595 Answers: 2 Comments: 0

y=(√(sin x+(√(sin x+(√(sin x+.............∞)))))) (dy/dx)=?

$$\mathrm{y}=\sqrt{\mathrm{sin}\:\mathrm{x}+\sqrt{\mathrm{sin}\:\mathrm{x}+\sqrt{\mathrm{sin}\:\mathrm{x}+.............\infty}}} \\ $$$$ \\ $$$$\frac{\mathrm{dy}}{\mathrm{dx}}=? \\ $$

Question Number 129489 Answers: 2 Comments: 0

if p(x+2)−2p(x)=x^2 −5x−3 find p(x)

$${if}\:{p}\left({x}+\mathrm{2}\right)−\mathrm{2}{p}\left({x}\right)={x}^{\mathrm{2}} −\mathrm{5}{x}−\mathrm{3} \\ $$$${find}\:{p}\left({x}\right) \\ $$

Question Number 129394 Answers: 3 Comments: 0

{ ((x^3 −3x^2 +5x+17=0)),((y^3 −3y^2 +5y+11=0)) :} find x+y .

$$\begin{cases}{\mathrm{x}^{\mathrm{3}} −\mathrm{3x}^{\mathrm{2}} +\mathrm{5x}+\mathrm{17}=\mathrm{0}}\\{\mathrm{y}^{\mathrm{3}} −\mathrm{3y}^{\mathrm{2}} +\mathrm{5y}+\mathrm{11}=\mathrm{0}}\end{cases} \\ $$$$\:\mathrm{find}\:\mathrm{x}+\mathrm{y}\:. \\ $$

Question Number 129364 Answers: 1 Comments: 0

Question Number 129343 Answers: 0 Comments: 1

f(x+1)=f(x−1)=x^2 f^(−1) (x)=??

$${f}\left({x}+\mathrm{1}\right)={f}\left({x}−\mathrm{1}\right)={x}^{\mathrm{2}} \:\:\:\:\:\:\:\:{f}^{−\mathrm{1}} \left({x}\right)=?? \\ $$

Question Number 129320 Answers: 1 Comments: 1

Question Number 129310 Answers: 0 Comments: 1

Question Number 129300 Answers: 2 Comments: 0

Find the polynomial divided by (x-2) the remaining 5 divided by (x-3) the remaining 9 and if divided by (x-4) the remaining 13 if divided by (x-10) 37 and if divided by (x-3/4) the remainder remains zero

$$ \\ $$Find the polynomial divided by (x-2) the remaining 5 divided by (x-3) the remaining 9 and if divided by (x-4) the remaining 13 if divided by (x-10) 37 and if divided by (x-3/4) the remainder remains zero

Question Number 129279 Answers: 0 Comments: 3

f(x)=((7x−3)/((2x−11)^(ln(1/7)) )) in which value of x is continuty ??

$${f}\left({x}\right)=\frac{\mathrm{7}{x}−\mathrm{3}}{\left(\mathrm{2}{x}−\mathrm{11}\right)^{{ln}\frac{\mathrm{1}}{\mathrm{7}}} }\:\:\:\:\:{in}\:{which}\:{value} \\ $$$${of}\:{x}\:{is}\:{continuty}\:?? \\ $$

Question Number 129255 Answers: 0 Comments: 1

x(1−(1/x))^x =?

$$\mathrm{x}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{x}}\right)^{\mathrm{x}} =? \\ $$

Question Number 129235 Answers: 2 Comments: 2

a= (7)^(1/3) + (9)^(1/3) b=4 Taqqoslang

$$\:\:\boldsymbol{{a}}=\:\sqrt[{\mathrm{3}}]{\mathrm{7}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{9}}\: \\ $$$$\:\:\:\boldsymbol{{b}}=\mathrm{4}\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{Taqqoslang}} \\ $$

Question Number 129148 Answers: 0 Comments: 1

[ ((4^x +44))^(1/5) + (1/( ((2^(x−1) +7))^(1/3) )) ]^8 = 336 x =?

$$\:\left[\:\sqrt[{\mathrm{5}}]{\mathrm{4}^{\mathrm{x}} +\mathrm{44}}\:+\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{3}}]{\mathrm{2}^{\mathrm{x}−\mathrm{1}} +\mathrm{7}}}\:\right]^{\mathrm{8}} =\:\mathrm{336} \\ $$$$\:\mathrm{x}\:=? \\ $$

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