Question

2x³+5x²-3x+5 Help plssss​

Answer 1

$f(x) = 2 {x}^{3} + 5 {x}^{2} - 3x + 5 = 0$

Nature of roots:

$Δ = \frac{4 {( {5}^{2} - 3(2)( - 3) )}^{3} - {(2 {(5})^{3} - (27) {2}^{2} (5) + 9(2)(5)( - 3))}^{2} }{27 \times {2}^{2} }$

$Δ = - 7459 < 0$

So,

$x_{1} \in ℝ$

Now, we can depress the cubic as following.

$f(x)≡ {x}^{3} + \frac{5}{2} {x}^{2} - \frac{3}{2} x + \frac{5}{2}$

$f_{x}≡ (x + \frac{5}{6} ) ^{3} - \frac{43}{12} (x + \frac{5}{6} ) + \frac{265}{54}$

$\implies x_{real} = - \frac{5}{6} + \sqrt[3]{( \frac{ - 265}{108}) + \sqrt{( \frac{265}{108} ) ^{2} - (\frac{43}{36} ) ^{3} \: } } + \sqrt[3]{( \frac{ - 265}{108} ) - \sqrt{(\frac{265}{108}) {}^{2} - ( \frac{43}{36} ) ^{3} \: } }$

Therefore, the three roots are :

$x _{1} = \frac{ - 5+ \sqrt[3]{ - 530 + 3 \sqrt{22377} } + \sqrt[3]{ - 530 - 3 \sqrt{22377} } }{6}$

$x_{2,3} = \frac{ - 10 + \sqrt[3]{530 + 3 \sqrt{22377} } + \sqrt[3]{530 - 3 \sqrt{22377} } + \sqrt{3} i( ± \sqrt[ 3]{ 530 + 3 \sqrt{22377} } ∓ \sqrt[3]{ 530 - 3 \sqrt{22377} } )}{12}$