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# Math Challenge #something

+1
308
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+333

I have lost track of these things...

Anyway, here is a problem that isn't that hard, really.

GIVEN:

a and b are real numbers

$$a+b=10$$

$$a^2+b^2=44$$

Find $$a^3+b^3$$

Oh, and last time, the question was declared answered after a moderator aswered it, so if possible, please don't declare this question answered so than people can solve it.

I encourage everyone who reads this to not look at the other people's answer, so that you can solve it yourself.

Mathhemathh  Aug 10, 2017
edited by Mathhemathh  Aug 10, 2017
edited by Mathhemathh  Aug 10, 2017

#3
+19207
+2

GIVEN:

a and b are real numbers

$$a+b = 10 \\ a^2+b^2 = 44$$

Find $$a^3 + b^3$$

$$\begin{array}{|rcll|} \hline (a+b)^2 &=& a^2 + 2ab + b^2 \\ (a+b)^2 &=& (a^2 + b^2) +2ab \\ (10)^2 &=& (44) + 2ab \\ 100 &=& 44 + 2ab \quad & | \quad : 2 \\ 50 &=& 22 + ab \\ ab &=& 50-22 \\ \mathbf{ab} & \mathbf{=} & \mathbf{28} \\\\ (a^2+b^2)(a+b) &=& a^3 +a^2b + b^2a + b^3 \\ (a^2+b^2)(a+b) &=& a^3 +b^3 + ab(a+b) \\ 44\cdot 10 &=& a^3 +b^3 + 28\cdot 10 \\ 440 &=& a^3 +b^3 + 280 \\ a^3 +b^3 &=& 440 - 280 \\ \mathbf{ a^3 +b^3} & \mathbf{=} & \mathbf{160} \\ \hline \end{array}$$

heureka  Aug 10, 2017
edited by heureka  Aug 10, 2017
Sort:

#1
0

a = 5 - i sqrt(3) ≈ 5.00000 - 1.73205 i and b = 5 + i sqrt(3) ≈ 5.00000 + 1.73205 i
a = 5 + i sqrt(3) ≈ 5.00000 + 1.73205 i and b = 5 - i sqrt(3) ≈ 5.00000 - 1.73205 i

Simplify the following:
(-(i sqrt(3)) + 5)^3 + (i sqrt(3) + 5)^3

(-(i sqrt(3)) + 5)^3 = (-(i sqrt(3)) + 5) (-(i sqrt(3)) + 5)^2:
(-(i sqrt(3)) + 5) (-(i sqrt(3)) + 5)^2 + (i sqrt(3) + 5)^3

(-(i sqrt(3)) + 5)^2 = 25 - 5 i sqrt(3) - 5 i sqrt(3) - 3 = 22 - 10 i sqrt(3):
(-(i sqrt(3)) + 5) -10 i sqrt(3) + 22 + (i sqrt(3) + 5)^3

(-i sqrt(3) + 5) (-10 i sqrt(3) + 22) = 5×22 + 5 (-10 i sqrt(3)) + -i sqrt(3)×22 + -i sqrt(3) (-10 i sqrt(3)) = 110 + -50 i sqrt(3) + -22 i sqrt(3) - 30 = -72 i sqrt(3) + 80:
-72 i sqrt(3) + 80 + (i sqrt(3) + 5)^3

(i sqrt(3) + 5)^3 = (i sqrt(3) + 5) (i sqrt(3) + 5)^2:
80 - 72 i sqrt(3) + (i sqrt(3) + 5) (i sqrt(3) + 5)^2

(i sqrt(3) + 5)^2 = 25 + 5 i sqrt(3) + 5 i sqrt(3) - 3 = 22 + 10 i sqrt(3):
80 - 72 i sqrt(3) + (i sqrt(3) + 5) 10 i sqrt(3) + 22

(i sqrt(3) + 5) (10 i sqrt(3) + 22) = 5×22 + 5×10 i sqrt(3) + i sqrt(3)×22 + i sqrt(3)×10 i sqrt(3) = 110 + 50 i sqrt(3) + 22 i sqrt(3) - 30 = 72 i sqrt(3) + 80:
80 - 72 i sqrt(3) + 72 i sqrt(3) + 80

80 - 72 i sqrt(3) + 80 + 72 i sqrt(3) = 160:

Guest Aug 10, 2017
#2
+333
+1

#mindblown

#overcomplicated

#i'mimpressed

I don't understand that behemoth of a solution because I was too lazy to read it, but you got the right answer, except there is a much, MUCH simpler solution.

Mathhemathh  Aug 10, 2017
#3
+19207
+2

GIVEN:

a and b are real numbers

$$a+b = 10 \\ a^2+b^2 = 44$$

Find $$a^3 + b^3$$

$$\begin{array}{|rcll|} \hline (a+b)^2 &=& a^2 + 2ab + b^2 \\ (a+b)^2 &=& (a^2 + b^2) +2ab \\ (10)^2 &=& (44) + 2ab \\ 100 &=& 44 + 2ab \quad & | \quad : 2 \\ 50 &=& 22 + ab \\ ab &=& 50-22 \\ \mathbf{ab} & \mathbf{=} & \mathbf{28} \\\\ (a^2+b^2)(a+b) &=& a^3 +a^2b + b^2a + b^3 \\ (a^2+b^2)(a+b) &=& a^3 +b^3 + ab(a+b) \\ 44\cdot 10 &=& a^3 +b^3 + 28\cdot 10 \\ 440 &=& a^3 +b^3 + 280 \\ a^3 +b^3 &=& 440 - 280 \\ \mathbf{ a^3 +b^3} & \mathbf{=} & \mathbf{160} \\ \hline \end{array}$$

heureka  Aug 10, 2017
edited by heureka  Aug 10, 2017
#4
+333
0

Good job.

Mathhemathh  Aug 11, 2017
#6
+473
0

UM HUMM UM UM HUMM

#5
+473
+1

a = 5 - i sqrt(3) ≈ 5.00000 - 1.73205 i and b = 5 + i sqrt(3) ≈ 5.00000 + 1.73205 i
a = 5 + i sqrt(3) ≈ 5.00000 + 1.73205 i and b = 5 - i sqrt(3) ≈ 5.00000 - 1.73205 i

Simplify the following:
(-(i sqrt(3)) + 5)^3 + (i sqrt(3) + 5)^3

(-(i sqrt(3)) + 5)^3 = (-(i sqrt(3)) + 5) (-(i sqrt(3)) + 5)^2:
(-(i sqrt(3)) + 5) (-(i sqrt(3)) + 5)^2 + (i sqrt(3) + 5)^3

(-(i sqrt(3)) + 5)^2 = 25 - 5 i sqrt(3) - 5 i sqrt(3) - 3 = 22 - 10 i sqrt(3):
(-(i sqrt(3)) + 5) -10 i sqrt(3) + 22 + (i sqrt(3) + 5)^3

(-i sqrt(3) + 5) (-10 i sqrt(3) + 22) = 5×22 + 5 (-10 i sqrt(3)) + -i sqrt(3)×22 + -i sqrt(3) (-10 i sqrt(3)) = 110 + -50 i sqrt(3) + -22 i sqrt(3) - 30 = -72 i sqrt(3) + 80:
-72 i sqrt(3) + 80 + (i sqrt(3) + 5)^3

(i sqrt(3) + 5)^3 = (i sqrt(3) + 5) (i sqrt(3) + 5)^2:
80 - 72 i sqrt(3) + (i sqrt(3) + 5) (i sqrt(3) + 5)^2

(i sqrt(3) + 5)^2 = 25 + 5 i sqrt(3) + 5 i sqrt(3) - 3 = 22 + 10 i sqrt(3):
80 - 72 i sqrt(3) + (i sqrt(3) + 5) 10 i sqrt(3) + 22

(i sqrt(3) + 5) (10 i sqrt(3) + 22) = 5×22 + 5×10 i sqrt(3) + i sqrt(3)×22 + i sqrt(3)×10 i sqrt(3) = 110 + 50 i sqrt(3) + 22 i sqrt(3) - 30 = 72 i sqrt(3) + 80:
80 - 72 i sqrt(3) + 72 i sqrt(3) + 80

80 - 72 i sqrt(3) + 80 + 72 i sqrt(3) = 160:                                      SO THE ANSWER IS 160