Problem 10 - A Graphical Approach to Precalculus with Limits

Question

Find all real solutions. Do not use a calculator. $$3 x^{3}+3 x=10 x^{2}$$

Step-by-Step Solution

Answer

The real solutions are \(x = 0\), \(x = 3\), and \(x = \frac{1}{3}\).

Step 1: Arrange the equation

Start by writing the given equation: \[3x^3 + 3x = 10x^2\] Our first task is to rearrange the terms to bring all expressions to one side of the equation so that it equals zero. Subtract \(10x^2\) from both sides:\[3x^3 + 3x - 10x^2 = 0\]

Step 2: Factor out the greatest common factor

Next, look for a common factor in all the terms on the left-side of the equation. Both \(3x^3\), \(10x^2\), and \(3x\) have at least \(x\) as a factor. Factor \(x\) out:\[x(3x^2 - 10x + 3) = 0\]

Step 3: Split into separate equations

Now that we have factored the equation, we can split it into two parts based on the factors. The equation is satisfied when either factor is zero:1. \(x = 0\)2. \(3x^2 - 10x + 3 = 0\)Solve each part separately.

Step 4: Solve the quadratic equation

Now solve the quadratic equation using the quadratic formula, where the quadratic formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a = 3\), \(b = -10\), and \(c = 3\). Plug these values into the formula:\[x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 3 \cdot 3}}{2 \cdot 3}\]\[x = \frac{10 \pm \sqrt{100 - 36}}{6}\]\[x = \frac{10 \pm \sqrt{64}}{6}\]\[x = \frac{10 \pm 8}{6}\]Calculate the two possible solutions for \(x\):- \(x = \frac{10 + 8}{6} = 3\)- \(x = \frac{10 - 8}{6} = \frac{1}{3}\)

Step 5: Write all solutions

Combine all solutions from the previous steps. The real solutions found are:1. \(x = 0\)2. \(x = 3\)3. \(x = \frac{1}{3}\)

Key Concepts

Factoring Polynomials

Factoring polynomials involves breaking down a polynomial into simpler "factor" polynomials that when multiplied together give back the original polynomial. This is a crucial skill in solving polynomial equations because it can simplify the equation significantly.

Consider the polynomial equation we started with:

• Equation: \(3x^3 + 3x = 10x^2\)

When rearranged and simplified, it becomes:

• Simplified: \(3x^3 + 3x - 10x^2 = 0\)

Here, we can factor out the greatest common factor (GCF) from all the terms, which is \(x\). By doing this early on, the equation becomes simpler to work with:

• Factored form: \(x(3x^2 - 10x + 3) = 0\)

Breaking polynomials into their factors makes it easier to split the equation into more manageable parts. This allows you to solve for the variable by setting each factor equal to zero.

Quadratic Formula

The quadratic formula is a universal method for finding solutions to quadratic equations, which are equations that can be put in the form \(ax^2 + bx + c = 0\). It is particularly useful when a quadratic cannot be easily factored.

For instance, in our example, once we isolated the quadratic part \(3x^2 - 10x + 3 = 0\), factoring it directly might be challenging, so we apply the quadratic formula:

• Formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)
• For this equation: \(a = 3\), \(b = -10\), \(c = 3\)

Plugging these values in, we find:

• \(x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 3 \cdot 3}}{2 \cdot 3}\)
• Simplifies to \(x = \frac{10 \pm \sqrt{100 - 36}}{6}\)
• Finally: \(x = \frac{10 \pm 8}{6}\)

This yields the solutions \(x = 3\) and \(x = \frac{1}{3}\), efficiently providing us the real roots of our quadratic part.

Real Solutions

Real solutions in an equation are the values of the variable that satisfy the equation under real number constraints. For polynomial equations, the solutions are the values of \(x\) that make the equation true—that is, when plugged into the equation, they produce an equality.

In our problem, by factoring and using the quadratic formula, we found three real solutions:

• \(x = 0\)
• \(x = 3\)
• \(x = \frac{1}{3}\)

These solutions correspond to the points where the graph of the polynomial would intersect the x-axis. Understanding which solutions are real is key in solving polynomial equations because not all polynomial solutions are real; some could be complex, which are not considered here.

Greatest Common Factor

The greatest common factor (GCF) of terms in an equation is the largest factor that divides each term without leaving a remainder. Finding the GCF is particularly useful for simplifying polynomial equations, as it allows you to factor out the common portion to make your equation more manageable.

In the equation we started with, the terms \(3x^3\), \(10x^2\), and \(3x\) all share a common factor of \(x\). Pulling out an \(x\) from each term:

• Factored form: \(x(3x^2 - 10x + 3) = 0\)

This reveals a simpler polynomial and reduces the complexity of solving it. Factoring out the GCF is a crucial step before any other form of simplification or solving. It reduces error chances and keeps the math straightforward.