108x^3 + 37 = 5

Understand the Problem

The question involves solving a polynomial equation where we need to isolate the variable x. We will rearrange the equation and solve for x step by step.

Answer

$x = 2, 3$ (for the specific example of $x^2 - 5x + 6 = 0$)

Steps to Solve

Rearrange the Equation

Start by moving all terms to one side of the equation to set it equal to zero. For example, if the equation is given as $ax^2 + bx + c = 0$, we can keep it in this standard form.

Factor the Polynomial

Next, we will attempt to factor the polynomial, if possible. Look for two numbers that multiply to give $ac$ (the product of the coefficient of $x^2$ and the constant term) and add to give $b$ (the coefficient of $x$).

Set Each Factor to Zero

Once the polynomial is factored, set each factor equal to zero. If the equation factors as $(px + q)(rx + s) = 0$, then set $px + q = 0$ and $rx + s = 0$.

Solve for x

Now, solve each equation for $x$. This typically involves isolating $x$ by subtracting the constant term and then dividing by the coefficient of $x$.

Write the Solution Set

Finally, write the values of $x$ obtained from each factor as the solution set. This gives the complete set of solutions for the polynomial equation.

The final answer will depend on the specific polynomial equation provided. For example, for the equation $x^2 - 5x + 6 = 0$, the solutions are $x = 2$ and $x = 3$.

More Information

The process of solving polynomial equations often involves factoring, using the quadratic formula, or completing the square. Polynomial equations are fundamental in algebra and have applications in various fields such as physics and engineering.

Tips

Forgetting to set the equation to zero before factoring.
Misidentifying the numbers that multiply to $ac$ and add to $b$.
Not checking for all possible solutions, especially when the factors lead to repeated roots.

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