5x^2-4x=5x+1 just need simplified you so much

Hey!

Using the additive property of equality, we can subtract 5x from both sides to get 5x^2-9x=1. Next, we can subtract 1 from both sides to get 5x^2-9x-1=0. Thus, we have a polynomial equation of the form ax^2+bx+c. To simplify this, we would need to factor it out. In the case where a>1, which this one is, we must factor the polynomial so we end up with (5x+ _)(x+_). Thus, if we put the first blank as the value a and the second value as the value b, we get (5x+a)*(x+b)=5x^2+ax+5xb+ab. Thus, a*b=-1 (since there isn’t an x attached to it) and ax+5xb=-9x. Using a*b=-1, we can divide b from both sides using the multiplicative property of equality to get a=-1/b. Substituting that into ax+5xb=-9x, we have ax+5(-1/a)x=-9x. Dividing x from both sides, we get a+-5/a=-9. Multiplying both sides by a get rid of the -5/a, we get a^2-5=-9a. Adding 9a to both sides, we end up with a^2+9a-5=0. This is simpler than our original equation, and we can quickly figure out that because (0)^2+9*0-5=-5 and (1)^2+9*1-5=5, a must be between 0 and 1 and it would therefore be relatively complex to factor out. Therefore, our equation stays as is, and our simplified equation is 5x^2-9x-1=0

Feel free to ask further questions, and have a great day!

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