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Recent problems solved by 'ikleyn'

ikleyn answered: 4 problems
Numeric_Fractions/479514: if a,b,c and d are in set of natural numbers(N) with a=bcd, b=cda and d=abc then,what is the value of (a+b+c+d)^2 ?

1 solutions
Answer 328557 by ikleyn(4) About Me  on 2011-08-09 00:30:07 (Show Source):
You can put this solution on YOUR website!
Since a=+bcd and b+=+cda, we can substitute b into the first equation and obtain a+=+%28c%2Ad%2Aa%29%2Ac%2Ad, which can be simplified into a+=+c%5E2%2Ad%5E2%2Aa. Since a,b,c,d are natural numbers (not equal to zero), we can divide both sides of the last equation by a and obtain 1+=+c%5E2%2Ad%5E2. Again, using the fact that a,b,c,d are natural numbers we conclude that both c and d are equal to 1.
Now we have to find values of a and b. We can use the last equation d+=+abc and substitute c = 1 and d = 1 to obtain 1+=+ab. Again, a and b are natural numbers, and thus both equal to 1. So, the only solution to the system of equations for a,b,c,d among the set of natural numbers is all of the being equal to 1.
Thus %28a+%2B+b+%2B+c+%2B+d%29%5E2 is equal to 16.

Rational-functions/478039: Simplify the rational expression
3x^2-7x+4/5x^2-8x+3
1 solutions
Answer 327581 by ikleyn(4) About Me  on 2011-08-04 22:35:07 (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • Here's what you tried:
  • Graphical form: %283x%5E2-7x%2B4%29%2F%285x%5E2-8x%2B3%29 simplifies to %283%2Ax-4%29%2F%285%2Ax-3%29
  • Text form: (3x^2-7x+4)/(5x^2-8x+3) simplifies to (3*x-4)/(5*x-3)
  • Cartoon (animation) form: simplify_cartoon%28+%283x%5E2-7x%2B4%29%2F%285x%5E2-8x%2B3%29+%29
    For tutors: simplify_cartoon( (3x^2-7x+4)/(5x^2-8x+3) )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at .
Canceled out common factors (x-(1))
Result: %283%2Ax-4%29%2F%285%2Ax-3%29

Universal Simplifier and Solver


Done!

test/444960: 2-10x=(3x-7)(x+1)
1 solutions
Answer 306623 by ikleyn(4) About Me  on 2011-05-04 13:26:27 (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • Here's what you tried:
  • This is an equation! Solutions: x=-3,x=1.
  • Graphical form: Equation 2-10x=%283x-7%29%2A%28x%2B1%29 was fully solved.
  • Text form: 2-10x=(3x-7)*(x+1) simplifies to 0=0
  • Cartoon (animation) form: simplify_cartoon%28+2-10x=%283x-7%29%2A%28x%2B1%29+%29
    For tutors: simplify_cartoon( 2-10x=(3x-7)*(x+1) )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at highlight_red%28+2+%29-10%2Ax=%283x-7%29%2A%28x%2B1%29.
Moved 2 to the right of expression
It becomes -10%2Ax%2Bhighlight_green%28+2+%29=%283x-7%29%2A%28x%2B1%29.
Look at -10%2Ax%2B2=highlight_red%28+%283%2Ax-7%29%2A%28x%2B1%29+%29.
Moved these terms to the left highlight_green%28+-%283%2Ax-7%29%2A%28x%2B1%29+%29
It becomes -10%2Ax%2B2-highlight_green%28+%283%2Ax-7%29%2A%28x%2B1%29+%29=0.
Look at -10%2Ax%2Bhighlight_red%28+2+%29-%283%2Ax-7%29%2A%28x%2B1%29=0.
Moved 2 to the right of expression
It becomes -10%2Ax-%283%2Ax-7%29%2A%28x%2B1%29%2Bhighlight_green%28+2+%29=0.
Look at -10%2Ax-highlight_red%28+%283%2Ax-7%29%2A%28x%2B1%29+%29%2B2=0.
Expanded term -%28x%2B1%29 by using associative property on %283%2Ax-7%29
It becomes .
Look at -10%2Ax-highlight_red%28+%28x%2B1%29%2A3%2Ax+%29%2B%28x%2B1%29%2A7%2B2=0.
Expanded term -3%2Ax by using associative property on %28x%2B1%29
It becomes .
Look at .
Multiplied numerator integers
It becomes -10%2Ax-3%2Ax%2Ax-highlight_green%28+3+%29%2Ax%2B%28x%2B1%29%2A7%2B2=0.
Look at -10%2Ax-3%2Ahighlight_red%28+x+%29%2Ahighlight_red%28+x+%29-3%2Ax%2B%28x%2B1%29%2A7%2B2=0.
Reduce similar several occurrences of highlight_red%28+x+%29 to highlight_green%28+x%5E2+%29
It becomes -10%2Ax-3%2Ahighlight_green%28+x%5E2+%29-3%2Ax%2B%28x%2B1%29%2A7%2B2=0.
Look at -highlight_red%28+10%2Ax+%29-3%2Ax%5E2-highlight_red%28+3%2Ax+%29%2B%28x%2B1%29%2A7%2B2=0.
Eliminated similar terms highlight_red%28+-10%2Ax+%29,highlight_red%28+-3%2Ax+%29 replacing them with highlight_green%28+%28-10-3%29%2Ax+%29
It becomes highlight_green%28+%28-10-3%29%2Ax+%29-3%2Ax%5E2%2B%28x%2B1%29%2A7%2B2=0.
Look at %28-highlight_red%28+10+%29-highlight_red%28+3+%29%29%2Ax-3%2Ax%5E2%2B%28x%2B1%29%2A7%2B2=0.
Added fractions or integers together
It becomes %28highlight_green%28+-13+%29%29%2Ax-3%2Ax%5E2%2B%28x%2B1%29%2A7%2B2=0.
Look at %28highlight_red%28+-13+%29%29%2Ax-3%2Ax%5E2%2B%28x%2B1%29%2A7%2B2=0.
Removed extra sign in front of -13
It becomes %28-highlight_green%28+13+%29%29%2Ax-3%2Ax%5E2%2B%28x%2B1%29%2A7%2B2=0.
Look at highlight_red%28+%28-highlight_red%28+13+%29%29%2Ax+%29-3%2Ax%5E2%2B%28x%2B1%29%2A7%2B2=0.
Remove unneeded parentheses around factor highlight_red%28+13+%29
It becomes -highlight_green%28+13+%29%2Ax-3%2Ax%5E2%2B%28x%2B1%29%2A7%2B2=0.
Look at -13%2Ax-3%2Ax%5E2%2Bhighlight_red%28+%28x%2B1%29%2A7+%29%2B2=0.
Expanded term 7 by using associative property on %28x%2B1%29
It becomes -13%2Ax-3%2Ax%5E2%2Bhighlight_green%28+7%2Ax+%29%2Bhighlight_green%28+7%2A1+%29%2B2=0.
Look at -13%2Ax-3%2Ax%5E2%2B7%2Ax%2Bhighlight_red%28+7+%29%2Ahighlight_red%28+1+%29%2B2=0.
Multiplied numerator integers
It becomes -13%2Ax-3%2Ax%5E2%2B7%2Ax%2Bhighlight_green%28+7+%29%2B2=0.
Look at -13%2Ax-3%2Ax%5E2%2B7%2Ax%2Bhighlight_red%28+7+%29%2Bhighlight_red%28+2+%29=0.
Added fractions or integers together
It becomes -13%2Ax-3%2Ax%5E2%2B7%2Ax%2Bhighlight_green%28+9+%29=0.
Look at -highlight_red%28+13%2Ax+%29-3%2Ax%5E2%2Bhighlight_red%28+7%2Ax+%29%2B9=0.
Eliminated similar terms highlight_red%28+-13%2Ax+%29,highlight_red%28+7%2Ax+%29 replacing them with highlight_green%28+%28-13%2B7%29%2Ax+%29
It becomes highlight_green%28+%28-13%2B7%29%2Ax+%29-3%2Ax%5E2%2B9=0.
Look at %28-highlight_red%28+13+%29%2Bhighlight_red%28+7+%29%29%2Ax-3%2Ax%5E2%2B9=0.
Added fractions or integers together
It becomes %28highlight_green%28+-6+%29%29%2Ax-3%2Ax%5E2%2B9=0.
Look at %28highlight_red%28+-6+%29%29%2Ax-3%2Ax%5E2%2B9=0.
Removed extra sign in front of -6
It becomes %28-highlight_green%28+6+%29%29%2Ax-3%2Ax%5E2%2B9=0.
Look at highlight_red%28+%28-highlight_red%28+6+%29%29%2Ax+%29-3%2Ax%5E2%2B9=0.
Remove unneeded parentheses around factor highlight_red%28+6+%29
It becomes -highlight_green%28+6+%29%2Ax-3%2Ax%5E2%2B9=0.
Look at highlight_red%28+-6%2Ax-3%2Ax%5E2%2B9+%29=0.
Equation highlight_red%28+-6%2Ax-3%2Ax%5E2%2B9=0+%29 is a quadratic equation: -3*x^2-6*x+9 =0, and has solutions -3,1
It becomes highlight_green%28+0+%29=0.
Result: 0=0
This is an equation! Solutions: x=-3,x=1.

Universal Simplifier and Solver


Done!

Equations/444963: solve this equation 2/3x=5/6
1 solutions
Answer 306620 by ikleyn(4) About Me  on 2011-05-04 13:21:17 (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: EXPLAIN simplification of an expression
Your Result:


YOUR ANSWER


  • Here's what you tried:
  • This is an equation! Solutions: x=1.25.
  • Graphical form: Equation 2%2F3%2Ax=5%2F6 was fully solved.
  • Text form: 2/3*x=5/6 simplifies to 0=0
  • Cartoon (animation) form: simplify_cartoon%28+2%2F3%2Ax=5%2F6+%29
    For tutors: simplify_cartoon( 2/3*x=5/6 )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

Look at 2%2F3%2Ax=highlight_red%28+5%2F6+%29.
Moved these terms to the left highlight_green%28+-5%2F6+%29
It becomes 2%2F3%2Ax-highlight_green%28+5%2F6+%29=0.
Look at highlight_red%28+2%2F3%2Ax-5%2F6+%29=0.
Solved linear equation highlight_red%28+2%2F3%2Ax-5%2F6=0+%29 equivalent to 0.666666666666667*x-0.833333333333333 =0
It becomes highlight_green%28+0+%29=0.
Result: 0=0
This is an equation! Solutions: x=1.25.

Universal Simplifier and Solver


Done!