How to find real number solutions?

Question

53.) x^2+8x+16=0

54.) x^2-4x+4=0

55.) x^3-15x^2+75x-125=0

56.) x^3+3x^2+3x+1=0

57.) 2x^2-x=3

58.) x^2+3x=10

59.) x^3+3x^2-2x-6=0

60.) x^4+7x^3-8x-56=0

61.) x^3+2x^2-x=2

62.) x^3-x^2+2x=2

Answer 1

The Rational Root Theorem says solutions exist that are factors of the constant term.

53) this is a perfect square, (x+4)^2=0, x+4=0, x=-4

54) same thing, (x-2)^2=0, x-2=0, x=2

55) try factors of 125, 5 works, then divide the equation by (x-5) for the others

56) -1 is a solution

57) bring the 3 to the left side, (2x-3)(x+1)=0, set each to 0 and get the solutions

58) bring the 10 to the left side, (x+5)(x-2)=0, ...

59) try factors of 6, -3 works, x=-3, divide (x+3) into the polynomial,...

60) x=2,...

61) x=1,...

62) x=1,...