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AlgebraQuestion and Answers: Page 247

Question Number 112053 Answers: 1 Comments: 1

solve for x, y, z ∈C: 2x^2 −3x=(√(13x^2 −52x+40)) 6y^2 −14x=(√(x^2 −220x+300)) z^2 −2z=(√(−12x^2 +72x−132)) [exact solutions possible in all cases]

$$\mathrm{solve}\:\mathrm{for}\:{x},\:{y},\:{z}\:\in\mathbb{C}: \\ $$$$\mathrm{2}{x}^{\mathrm{2}} −\mathrm{3}{x}=\sqrt{\mathrm{13}{x}^{\mathrm{2}} −\mathrm{52}{x}+\mathrm{40}} \\ $$$$\mathrm{6}{y}^{\mathrm{2}} −\mathrm{14}{x}=\sqrt{{x}^{\mathrm{2}} −\mathrm{220}{x}+\mathrm{300}} \\ $$$${z}^{\mathrm{2}} −\mathrm{2}{z}=\sqrt{−\mathrm{12}{x}^{\mathrm{2}} +\mathrm{72}{x}−\mathrm{132}} \\ $$$$\left[\mathrm{exact}\:\mathrm{solutions}\:\mathrm{possible}\:\mathrm{in}\:\mathrm{all}\:\mathrm{cases}\right] \\ $$

Question Number 112008 Answers: 0 Comments: 0

Question Number 111939 Answers: 2 Comments: 0

Question Number 111982 Answers: 2 Comments: 0

Question Number 111909 Answers: 2 Comments: 0

solve { ((x(√x)+y(√y) = 5)),((x(√y) +y(√x) = 1)) :}

$${solve}\:\begin{cases}{{x}\sqrt{{x}}+{y}\sqrt{{y}}\:=\:\mathrm{5}}\\{{x}\sqrt{{y}}\:+{y}\sqrt{{x}}\:=\:\mathrm{1}}\end{cases} \\ $$

Question Number 111724 Answers: 0 Comments: 6

How many real numbers x satisfy the equation 3^(2x+2) −3^(x+3) −3^x +3=0 ?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{x}\:\mathrm{satisfy}\:\mathrm{the} \\ $$$$\mathrm{equation}\:\mathrm{3}^{\mathrm{2x}+\mathrm{2}} −\mathrm{3}^{\mathrm{x}+\mathrm{3}} −\mathrm{3}^{\mathrm{x}} +\mathrm{3}=\mathrm{0}\:? \\ $$

Question Number 111623 Answers: 1 Comments: 2

(x−2)(x+3)(x−1)^2 ≥ 0

$$\left({x}−\mathrm{2}\right)\left({x}+\mathrm{3}\right)\left({x}−\mathrm{1}\right)^{\mathrm{2}} \:\geqslant\:\mathrm{0} \\ $$

Question Number 111601 Answers: 1 Comments: 0

Question Number 111586 Answers: 0 Comments: 3

DTM−2020 savollaridan biri. ∫x^x = ? t.me/matematik_olimpiadachilar

$$\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{DTM}}−\mathrm{2020}\:\:\boldsymbol{{savollaridan}}\:\:\boldsymbol{{biri}}. \\ $$$$\:\int\boldsymbol{{x}}^{\boldsymbol{{x}}} =\:? \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{{t}}.\boldsymbol{{me}}/\boldsymbol{{matematik\_olimpiadachilar}} \\ $$

Question Number 111583 Answers: 1 Comments: 0

without using a substitution for 3^x solve the equation 9^x −3^(x + 2) −36 = 0

$$\mathrm{without}\:\mathrm{using}\:\mathrm{a}\:\mathrm{substitution}\:\mathrm{for}\:\mathrm{3}^{{x}} \:\mathrm{solve}\:\mathrm{the}\:\mathrm{equation} \\ $$$$\:\:\mathrm{9}^{{x}} −\mathrm{3}^{{x}\:+\:\mathrm{2}} −\mathrm{36}\:=\:\mathrm{0} \\ $$

Question Number 111560 Answers: 2 Comments: 1

Question Number 111534 Answers: 2 Comments: 0

If x=((1+(√(2016)))/2), then 4x^3 −2019x−2017 equals?

$$\mathrm{If}\:\mathrm{x}=\frac{\mathrm{1}+\sqrt{\mathrm{2016}}}{\mathrm{2}},\:\mathrm{then} \\ $$$$\mathrm{4x}^{\mathrm{3}} −\mathrm{2019x}−\mathrm{2017}\:\mathrm{equals}? \\ $$

Question Number 111365 Answers: 0 Comments: 0

Question Number 111344 Answers: 0 Comments: 3

(√3)!=?

$$\sqrt{\mathrm{3}}!=? \\ $$

Question Number 111343 Answers: 0 Comments: 1

i!=?

$${i}!=? \\ $$

Question Number 111208 Answers: 1 Comments: 3

let p a prime number s.t p≥7 and a=333......3_(p−1 times) Show that 11∣a.

$$\:\mathrm{let}\:{p}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number}\:\mathrm{s}.\mathrm{t}\:{p}\geqslant\mathrm{7}\:\mathrm{and}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{a}=\underset{{p}−\mathrm{1}\:{times}} {\mathrm{333}......\mathrm{3}} \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:\mathrm{11}\mid{a}. \\ $$

Question Number 111175 Answers: 1 Comments: 0

Question Number 111134 Answers: 0 Comments: 0

Question Number 111062 Answers: 0 Comments: 2

Question Number 111029 Answers: 1 Comments: 0

Question Number 111008 Answers: 0 Comments: 3

((√(x+1))/(y+2)) + ((√(y+2))/(x+1)) =1 => x=?

$$\frac{\sqrt{\boldsymbol{{x}}+\mathrm{1}}}{\boldsymbol{{y}}+\mathrm{2}}\:+\:\frac{\sqrt{\boldsymbol{{y}}+\mathrm{2}}}{\boldsymbol{{x}}+\mathrm{1}}\:=\mathrm{1}\:\:\:\:\:\:=>\:\:\boldsymbol{{x}}=? \\ $$

Question Number 110897 Answers: 3 Comments: 0

(1)4x−4 ≤ ∣x^2 −3x+2 ∣ find the solution set (2) ((1+cos ((α/2))−sin ((α/2)))/(1−cos ((α/2))−sin ((α/2))))=?

$$\left(\mathrm{1}\right)\mathrm{4x}−\mathrm{4}\:\leqslant\:\mid\mathrm{x}^{\mathrm{2}} −\mathrm{3x}+\mathrm{2}\:\mid\: \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{solution}\:\mathrm{set}\: \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}+\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)}{\mathrm{1}−\mathrm{cos}\:\left(\frac{\alpha}{\mathrm{2}}\right)−\mathrm{sin}\:\left(\frac{\alpha}{\mathrm{2}}\right)}=? \\ $$

Question Number 110879 Answers: 0 Comments: 1

Question Number 110843 Answers: 2 Comments: 0

(x+1)^((x+1)) =(√2) find all values of x (Please step by step)

$$\left({x}+\mathrm{1}\right)^{\left({x}+\mathrm{1}\right)} =\sqrt{\mathrm{2}}\:\:\:\:\:\:{find}\:{all}\:{values}\:{of}\:{x} \\ $$$$\left({Please}\:{step}\:{by}\:{step}\right) \\ $$

Question Number 110837 Answers: 0 Comments: 0

Question Number 110799 Answers: 0 Comments: 0

What is the he minimum value of (√(x^2 +1))+(√((y−x)^2 +25))+(√((z−y)^2 +4))+(√((9−z)^2 +16)) if (x, y and z) ∈ R

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{What}\:{is}\:{the}\:{he}\:{minimum}\:{value}\:{of}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}+\sqrt{\left({y}−{x}\right)^{\mathrm{2}} +\mathrm{25}}+\sqrt{\left({z}−{y}\right)^{\mathrm{2}} +\mathrm{4}}+\sqrt{\left(\mathrm{9}−{z}\right)^{\mathrm{2}} +\mathrm{16}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{if}\:\:\left({x},\:{y}\:{and}\:{z}\right)\:\in\:\mathbb{R}\: \\ $$

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