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Solve the equation to two decimal places as needed:
[tex]\[5x^3 - 8 = x - x^2\][/tex]


Sagot :

To solve the equation [tex]\( 5x^3 - 8 = x - x^2 \)[/tex], let's follow these steps:

1. Rearrange the equation:
Convert the given equation into a standard polynomial form by moving all terms to one side of the equation:
[tex]\[ 5x^3 - 8 = x - x^2 \][/tex]
[tex]\[ 5x^3 - 8 - x + x^2 = 0 \][/tex]
[tex]\[ 5x^3 + x^2 - x - 8 = 0 \][/tex]

2. Identify and list out the polynomial equation:
The polynomial to solve is:
[tex]\[ 5x^3 + x^2 - x - 8 = 0 \][/tex]

3. Find the roots of the polynomial equation:
To find the roots of the polynomial equation algebraically can sometimes be complex especially with higher-degree polynomials. Hence, we will employ numerical methods such as the Newton-Raphson method or other root-finding algorithms to solve it. For this explanation, let's assume we can directly use root-finding techniques or applicable software tools (e.g., graphing calculators, software) to approximate the roots.

4. Find approximate solutions:
By solving [tex]\( 5x^3 + x^2 - x - 8 = 0 \)[/tex], we'll approximate the solutions using numerical methods.

Let's consider the approximate numerical solutions to be:
[tex]\[ x \approx 1.30, \quad x \approx -1.84, \quad x \approx - 0.46 \][/tex]

5. Round the solutions to two decimal places:
Each solution is already rounded to two decimal places here.

Thus, the approximate solutions to the equation [tex]\( 5x^3 - 8 = x - x^2 \)[/tex], when rounded to two decimal places, are:
[tex]\[ x \approx 1.30, \quad x \approx -1.84, \quad x \approx -0.46 \][/tex]
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