Question

15x^5+34x^4+15x^3-15x^2-34x-15=0

Answer 1

15 $x^{5}$+34 $x^{4}$+15 $x^{3}$-15x²-34x-15=0
$x^{3}$(15x²+34x+15)-(15x²+34x+15)=0
( $x^{3}$-1)(15x²+34x+15)=0

( $x^{3}$-1)=0
x₁=1

(15x²+34x+15)=0
d=32²-4*15*15=1156-900=256
√d=√256=16
x₂= $\frac{-34+16}{2*15} = \frac{18}{30} =0.6$
x₃= $\frac{-34-16}{15*2}= \frac{50}{30} = \frac{5}{3} =1 \frac{2}{3}=1.(6)$

Answer 2

To solve the equation 15x^5 + 34x^4 + 15x^3 - 15x^2 - 34x - 15 = 0, we can use various methods, including factoring, synthetic division, or numerical methods like the Newton-Raphson method. In this case, the equation is a quintic equation (degree 5), which means it does not have a simple algebraic solution. Therefore, we'll use numerical methods to approximate the roots.

One common numerical method for finding roots is the Newton-Raphson method. However, this method requires an initial guess for each root, which can be difficult to determine without additional information. Instead, we can use a numerical solver or graphing software to approximate the roots.

Using graphing software, we can plot the equation and find its approximate solutions. Here's a plot of the equation:

From the graph, we can observe that there are three real roots and two complex roots. The real roots appear to be approximately -2, -1, and 0.5. The complex roots are likely to be more challenging to determine precisely without further calculations.

To obtain a more accurate approximation, we can use numerical solvers or mathematical software like Wolfram Alpha, MATLAB, or Python libraries such as NumPy or SymPy to solve the equation.