Question 920793
i believe you meant ^ rather than *


* is the symbol used for multiplication.
^ is the symbol used for exponentiation.


the way you showed what you did leads me to believe you meant ^ rather than *.


i'll assume that and see where it goes.


start with (2x^2 * 5x) - (3x * 4x) = 32


simplify to get:


10x^3 - 12x^2 = 32


subtract 32 from both sides of the equation to get:


10x^3 - 12x^2 - 32 = 0


being lazy, i didn't go to the trouble of finding the factors of this equation.


i just used my calculator to graph it and i got x = 2 as the solution.


the graph is shown below:


{{{graph(600,600,-5,5,-100,100,10x^3 - 12x^2 - 32)}}}


if you were to do this the hard way, you would have had to find the factors of 32 divided by the factors of 10 and then tested each one to see if any of them lead to a rational solution using synthetic algebra.


the possible factors of 32 are 32, 16, 8, 4, 2, 1 


the possible factors of 10 are 10, 5, 2, 1


the possible factors of 32 divided by the possible factors of 10 are:


plus or minus {32, 16, 8, 4, 2, 1, 32/5, 16/5, 8/5, 4/5, 2/5, 1/5, 1/10}


you would need to test each one in turn.


i'll be lazy again and jump to the correct solution of 2.


using synthetic division, you would divide 2 into the coefficients of 10x^3 -12x^2 - 32.


you would first need to add the x term because all the different orders in the equation need to be shown.


you will get:


2 divided into the coefficients of 10x^3 - 12x^2 - 0x - 32


that would be 2 divided into 10 - 12 - 0 - 32


using synthetic division you would get the solution of:


10 + 8 + 16 + 0


that equates to an equation of 10x^2 + 8x + 16 with a remainder of 0.


now you want to solve the quadratic equation of 10x^2 + 8x + 16 = 0


divide both sides of that equation by 2 to get 5x^2 + 4x + 8 = 0


that equation doesn't factor, so use the quadratic formula.


since b^2 - 4ac is negative, this quadratic equation doesn't have any real roots.


you are left with the only real root of the cubic equation being x = 2.


the graph confirms that.


your solution is that x = 2.


if you need tutorials on how to do synthetic division or how to factor using the quadratic formula, let me know and i'll send you links that teach you how to do them.


i can also send you links on how to find the rational factors of a polynomial equation if you need that as well.