1) Let's simplify the first equation step by step:
(x²-x)² - 18(x²-x-2) +36 = 0Expand the square term:(x²-x)(x²-x) - 18(x²-x-2) +36 = 0(x^4 - 2x^3 + x^2) - 18(x^2 - x - 2) + 36 = 0x^4 - 2x^3 + x^2 - 18x^2 + 18x + 36 + 36 = 0x^4 - 2x^3 - 17x^2 + 18x + 72 = 0
Therefore, the simplified form of the first equation is x^4 - 2x^3 - 17x^2 + 18x + 72 = 0.
2) Let's simplify the second equation:
√(x² + 3x - 10) = -x² - 3x + 10Square both sides to eliminate the square root:(x² + 3x - 10) = (-x² - 3x + 10)^2(x² + 3x - 10) = (x^4 + 6x^3 + 13x^2 - 20x + 100)
Expand the right side:x² + 3x - 10 = x^4 + 6x^3 + 13x^2 - 20x + 100
Rearrange the terms to get a polynomial equation in standard form:x^4 + 6x^3 + 13x^2 - 20x + 100 - x^2 - 3x + 10 = 0x^4 + 6x^3 + 12x^2 - 23x + 90 = 0
Therefore, the simplified form of the second equation is x^4 + 6x^3 + 12x^2 - 23x + 90 = 0.
1) Let's simplify the first equation step by step:
(x²-x)² - 18(x²-x-2) +36 = 0
Expand the square term:
(x²-x)(x²-x) - 18(x²-x-2) +36 = 0
(x^4 - 2x^3 + x^2) - 18(x^2 - x - 2) + 36 = 0
x^4 - 2x^3 + x^2 - 18x^2 + 18x + 36 + 36 = 0
x^4 - 2x^3 - 17x^2 + 18x + 72 = 0
Therefore, the simplified form of the first equation is x^4 - 2x^3 - 17x^2 + 18x + 72 = 0.
2) Let's simplify the second equation:
√(x² + 3x - 10) = -x² - 3x + 10
Square both sides to eliminate the square root:
(x² + 3x - 10) = (-x² - 3x + 10)^2
(x² + 3x - 10) = (x^4 + 6x^3 + 13x^2 - 20x + 100)
Expand the right side:
x² + 3x - 10 = x^4 + 6x^3 + 13x^2 - 20x + 100
Rearrange the terms to get a polynomial equation in standard form:
x^4 + 6x^3 + 13x^2 - 20x + 100 - x^2 - 3x + 10 = 0
x^4 + 6x^3 + 12x^2 - 23x + 90 = 0
Therefore, the simplified form of the second equation is x^4 + 6x^3 + 12x^2 - 23x + 90 = 0.