A quartic equation is a polynomial equation of degree four. It's the most complex equation that can be solved by radicals according to the Abel-Ruffini theorem.
The general form of a quartic equation is:
\[ ax^4 + bx^3 + cx^2 + dx + e = 0 \]
Where:
\[ x^4 - 10x^3 + 35x^2 - 50x + 24 = 0 \]
| \( a = 1 \) | \( b = -10 \) | \( c = 35 \) | \( d = -50 \) | \( e = 24 \) |
First, we depress the quartic by substituting \( x = y - \frac{b}{4a} \):
\[ y = x + \frac{10}{4} = x + \frac{5}{2} \]
This gives us the depressed quartic:
\[ y^4 + py^2 + qy + r = 0 \]
Where:
\[ p = -\frac{3b^2}{8a^2} + \frac{c}{a} = -\frac{3(100)}{8} + 35 = -37.5 + 35 = -2.5 \]
\[ q = \frac{b^3}{8a^3} - \frac{bc}{2a^2} + \frac{d}{a} = \frac{-1000}{8} - 175 - 50 = -350 \]
\[ r = -\frac{3b^4}{256a^4} + \frac{b^2c}{16a^3} - \frac{bd}{4a^2} + \frac{e}{a} = -\frac{30000}{256} + \frac{3500}{16} - \frac{500}{4} + 24 = 15.625 \]
The roots are:
\[ (4)^4 - 10(4)^3 + 35(4)^2 - 50(4) + 24 = 256 - 640 + 560 - 200 + 24 = 0 \]
\[ (3)^4 - 10(3)^3 + 35(3)^2 - 50(3) + 24 = 81 - 270 + 315 - 150 + 24 = 0 \]
\[ (2)^4 - 10(2)^3 + 35(2)^2 - 50(2) + 24 = 16 - 80 + 140 - 100 + 24 = 0 \]
\[ (1)^4 - 10(1)^3 + 35(1)^2 - 50(1) + 24 = 1 - 10 + 35 - 50 + 24 = 0 \]
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