Answer :
Sure! Let's solve each of these equations one by one:
19. Solve the equation:
[tex]\[ -3x^3 - x^2 + 54x - 40 = 2x^2 + 6x + 20 \][/tex]
First, move all the terms to one side of the equation:
[tex]\[ -3x^3 - x^2 + 54x - 40 - 2x^2 - 6x - 20 = 0 \][/tex]
Combine like terms:
[tex]\[ -3x^3 - 3x^2 + 48x - 60 = 0 \][/tex]
The solutions to this equation are [tex]\( x = -5 \)[/tex] and [tex]\( x = 2 \)[/tex].
20. Solve the equation:
[tex]\[ 2x^3 + 3x^2 - 36 = x^3 - x^2 + 9x \][/tex]
First, move all the terms to one side of the equation:
[tex]\[ 2x^3 + 3x^2 - 36 - x^3 + x^2 - 9x = 0 \][/tex]
Combine like terms:
[tex]\[ x^3 + 4x^2 - 9x - 36 = 0 \][/tex]
The solutions to this equation are [tex]\( x = -4 \)[/tex], [tex]\( x = -3 \)[/tex], and [tex]\( x = 3 \)[/tex].
21. Solve the equation:
[tex]\[ -5x^4 + 4x^2 - 12x = -6x^4 + 3x^3 \][/tex]
First, move all the terms to one side of the equation:
[tex]\[ -5x^4 + 4x^2 - 12x + 6x^4 - 3x^3 = 0 \][/tex]
Combine like terms:
[tex]\[ x^4 - 3x^3 + 4x^2 - 12x = 0 \][/tex]
The solutions to this equation are [tex]\( x = 0 \)[/tex], [tex]\( x = 3 \)[/tex], [tex]\( x = -2i \)[/tex], and [tex]\( x = 2i \)[/tex].
Remember, these solutions could include complex numbers if real solutions don't completely satisfy the equation.
19. Solve the equation:
[tex]\[ -3x^3 - x^2 + 54x - 40 = 2x^2 + 6x + 20 \][/tex]
First, move all the terms to one side of the equation:
[tex]\[ -3x^3 - x^2 + 54x - 40 - 2x^2 - 6x - 20 = 0 \][/tex]
Combine like terms:
[tex]\[ -3x^3 - 3x^2 + 48x - 60 = 0 \][/tex]
The solutions to this equation are [tex]\( x = -5 \)[/tex] and [tex]\( x = 2 \)[/tex].
20. Solve the equation:
[tex]\[ 2x^3 + 3x^2 - 36 = x^3 - x^2 + 9x \][/tex]
First, move all the terms to one side of the equation:
[tex]\[ 2x^3 + 3x^2 - 36 - x^3 + x^2 - 9x = 0 \][/tex]
Combine like terms:
[tex]\[ x^3 + 4x^2 - 9x - 36 = 0 \][/tex]
The solutions to this equation are [tex]\( x = -4 \)[/tex], [tex]\( x = -3 \)[/tex], and [tex]\( x = 3 \)[/tex].
21. Solve the equation:
[tex]\[ -5x^4 + 4x^2 - 12x = -6x^4 + 3x^3 \][/tex]
First, move all the terms to one side of the equation:
[tex]\[ -5x^4 + 4x^2 - 12x + 6x^4 - 3x^3 = 0 \][/tex]
Combine like terms:
[tex]\[ x^4 - 3x^3 + 4x^2 - 12x = 0 \][/tex]
The solutions to this equation are [tex]\( x = 0 \)[/tex], [tex]\( x = 3 \)[/tex], [tex]\( x = -2i \)[/tex], and [tex]\( x = 2i \)[/tex].
Remember, these solutions could include complex numbers if real solutions don't completely satisfy the equation.
