I've got nothing interesting to do right now so I'll just post these. Find the answer or simplify/evaluate the expression.
No. 6:
Solve for x over the real numbers:
8/x = x + 1/x
Bring x + 1/x together using the common denominator x:
8/x = (x^2 + 1)/x
Multiply both sides by x:
8 = x^2 + 1
8 = x^2 + 1 is equivalent to x^2 + 1 = 8:
x^2 + 1 = 8
Subtract 1 from both sides:
x^2 = 7
Take the square root of both sides:
Answer: | x = sqrt(7) or x = -sqrt(7)
No.7:
x/y = 1/(16 (x^2)^(5/3))
No.5:
Simplify the following:
sqrt(10000) - sqrt(81) + sqrt(81)
sqrt(10000) - sqrt(81) + sqrt(81) = 100:
Answer: | 100
No.4:
Solve for x:
x^4 + 3 x^2 + 3 = 1
Subtract 1 from both sides:
x^4 + 3 x^2 + 2 = 0
Substitute y = x^2:
y^2 + 3 y + 2 = 0
The left hand side factors into a product with two terms:
(y + 1) (y + 2) = 0
Split into two equations:
y + 1 = 0 or y + 2 = 0
Subtract 1 from both sides:
y = -1 or y + 2 = 0
Substitute back for y = x^2:
x^2 = -1 or y + 2 = 0
Take the square root of both sides:
x = i or x = -i or y + 2 = 0
Subtract 2 from both sides:
x = i or x = -i or y = -2
Substitute back for y = x^2:
x = i or x = -i or x^2 = -2
Take the square root of both sides:
Answer: | x = i or x = -i or x = i sqrt(2) or x= -i sqrt(2)
No.3:
Expand the following:
(3^(1/3) + 2^(1/3)) (4^(1/3) - 6^(1/3) + 9^(1/3))
4^(1/3) = (2^2)^(1/3):
(3^(1/3) + 2^(1/3)) (2^(2/3) - 6^(1/3) + 9^(1/3))
9^(1/3) = (3^2)^(1/3):
(3^(1/3) + 2^(1/3)) (2^(2/3) - 6^(1/3) + 3^(2/3))
(2^(2/3) + 3^(2/3) - 6^(1/3)) (2^(1/3) + 3^(1/3)) = 2^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3)) + 3^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3)):
2^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3)) + 3^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3))
2^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3)) = 2^(1/3)×2^(2/3) + 2^(1/3)×3^(2/3) + 2^(1/3) (-6^(1/3)):
2^(1/3)×2^(2/3) + 2^(1/3)×3^(2/3) - 2^(1/3) 6^(1/3) + 3^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3))
2^(1/3)×2^(2/3) = 2^(1/3 + 2/3):
2^(1/3 + 2/3) + 2^(1/3)×3^(2/3) - 2^(1/3) 6^(1/3) + 3^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3))
2^(1/3 + 2/3) = 2:
2 + 2^(1/3)×3^(2/3) - 2^(1/3) 6^(1/3) + 3^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3))
2^(1/3) 6^(1/3) = (2×6)^(1/3):
2 + 2^(1/3)×3^(2/3) - (2×6)^(1/3) + 3^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3))
2×6 = 12:
2 + 2^(1/3)×3^(2/3) - (12 )^(1/3) + 3^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3))
12^(1/3) = (2^2×3)^(1/3) = 2^(2/3) 3^(1/3):
2 + 2^(1/3)×3^(2/3) - 2^(2/3) 3^(1/3) + 3^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3))
3^(1/3) (2^(2/3) + 3^(2/3) - 6^(1/3)) = 3^(1/3)×2^(2/3) + 3^(1/3)×3^(2/3) + 3^(1/3) (-6^(1/3)):
2 + 2^(1/3)×3^(2/3) - 2^(2/3) 3^(1/3) + 3^(1/3)×2^(2/3) + 3^(1/3)×3^(2/3) - 3^(1/3) 6^(1/3)
3^(1/3)×3^(2/3) = 3^(1/3 + 2/3):
2 + 2^(1/3)×3^(2/3) - 2^(2/3) 3^(1/3) + 3^(1/3)×2^(2/3) + 3^(1/3 + 2/3) - 3^(1/3) 6^(1/3)
3^(1/3 + 2/3) = 3:
2 + 2^(1/3)×3^(2/3) - 2^(2/3) 3^(1/3) + 3^(1/3)×2^(2/3) + 3 - 3^(1/3) 6^(1/3)
3^(1/3) 6^(1/3) = (3×6)^(1/3):
2 + 2^(1/3)×3^(2/3) - 2^(2/3) 3^(1/3) + 3^(1/3)×2^(2/3) + 3 - (3×6)^(1/3)
3×6 = 18:
2 + 2^(1/3)×3^(2/3) - 2^(2/3) 3^(1/3) + 3^(1/3)×2^(2/3) + 3 - (18 )^(1/3)
18^(1/3) = (2×3^2)^(1/3) = 2^(1/3)×3^(2/3):
2 + 2^(1/3)×3^(2/3) - 2^(2/3) 3^(1/3) + 3^(1/3)×2^(2/3) + 3 - 2^(1/3)×3^(2/3)
2 + 2^(1/3)×3^(2/3) - 2^(2/3) 3^(1/3) + 3^(1/3)×2^(2/3) + 3 - 2^(1/3)×3^(2/3) = 5:
Answer: | 5
No.2:
Solve for x:
sqrt(x + 5) - 2 = 6
Add 2 to both sides:
sqrt(x + 5) = 8
Raise both sides to the power of two:
x + 5 = 64
Subtract 5 from both sides:
Answer: | x = 59
No.1:
Solve for x:
x^2 = x - 1
Subtract x - 1 from both sides:
x^2 - x + 1 = 0
Subtract 1 from both sides:
x^2 - x = -1
Add 1/4 to both sides:
x^2 - x + 1/4 = -3/4
Write the left hand side as a square:
(x - 1/2)^2 = -3/4
Take the square root of both sides:
x - 1/2 = (i sqrt(3))/2 or x - 1/2 = 1/2 (-i) sqrt(3)
Add 1/2 to both sides:
x = 1/2 + (i sqrt(3))/2 or x - 1/2 = 1/2 (-i) sqrt(3)
Add 1/2 to both sides:
Answer: | x = 1/2 + (i sqrt(3))/2 or x = 1/2 - (i sqrt(3))/2. No "Real Solutions" !!!!
1. | x2 = x - 1 | ||
x2 - x + 1 = 0 | |||
x = ( 1 ± √[ 1 - 4] )/ 2 | |||
x = ( 1 ± √3 i ) / 2 | There are no real solutions for x . | ||
2. | √[ x + 5] - 2 = 6 | ||
√[ x + 5] = 8 | |||
x + 5 = 82 | |||
x = 82 - 5 | |||
x = 59 |
3. (3√3+3√2)(3√4−3√6+3√9)=(3√3)(3√4)−(3√3)(3√6)+(3√3)(3√9)+(3√2)(3√4)−(3√2)(3√6)+(3√2)(3√9)=3√12−3√18+3√27+3√8−3√12+3√18=3√27+3√8=3+2=5
5. √10000−√81+√81=√10000=√104=102=100
6. 8x=x+1x 8=x2+1 7=x2x=±√7 The smallest value is -sqrt(7) .
Let's let everyone jump in! Since hecticlar didn't do #4, I will do number 4.
x4+3x2+3=1
x4+3x2+3=1 | Subtract 1 on both sides. | ||
x4+3x2+2=0 | This expression is factorable. Think: What number multiplies to get 2 and add to get 3. That's right! 2 and 1! | ||
(x2+1)(x2+2)=0 | Set both factors equal to 0 and solve. | ||
| |||
| Take the square root of both sides. | ||
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Now, let's simplify both solutions:
x=±√−1=±i
x=±√−2=±√−1∗2=±i√2
Those are your solutions all done and dusted!
Note that for (1) we can rearrange it as
x^2 - x + 1 = 0
The discriminant for this is b^2 - 4ac = (-1)^2 - 4 (1) (1) = -3
And when the discriminant < 0, we will have no real solutions
Gonna solve that all :P
1)
x2=x−1x2−x+1=0Δ=b2−4ac=(−1)2−4(1)(1)=−3∴No real solutions.
2)
√x+5−2=6√x+5=8x+5=64x=59
3)
Note that:a3+b3=(a+b)(a2−ab+b2)(3√3+3√2)(3√4−3√6+3√9)=(3√3+3√2)((3√2)2−(3√2)(3√3)+(3√3)2)=(3√3)3+(3√2)3=3+2=5
4)
x4+3x2+3=1|u=x2u2+3u+2=0(u+1)(u+2)=0x2+1=0 or x2+2=0x=i,x=−i,x=√2i,x=−√2i
5)
√10000−√81+√81=√10000=100
6)
8x=x+1x8=x2+1x2−7=0x=√7 or x=−√7
7) Not enough information...