Problem 62 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ 3 x(3 x+10)=6 x-16 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = -\frac{4}{3}\).

Step 1: Distribute and Simplify

Let's begin by expanding the equation. We have to distribute the terms in the brackets: \(3x(3x + 10) = 6x - 16\). This gives us:\[ 3x \cdot 3x + 3x \cdot 10 = 6x - 16 \]Simplifying the left side, we get:\[ 9x^2 + 30x = 6x - 16 \]

Step 2: Move All Terms to One Side

Next, let's move all terms to one side to set the equation to zero:\[ 9x^2 + 30x - 6x + 16 = 0 \]This further simplifies to:\[ 9x^2 + 24x + 16 = 0\]

Step 3: Use The Quadratic Formula

We now have a quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 9\), \(b = 24\), and \(c = 16\). We'll use the quadratic formula:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substituting the values in, we get:\[ x = \frac{-24 \pm \sqrt{24^2 - 4 \times 9 \times 16}}{2 \times 9} \]This simplifies to:\[ x = \frac{-24 \pm \sqrt{576 - 576}}{18} \]\[ x = \frac{-24 \pm \sqrt{0}}{18} \]

Step 4: Solve for x

Since \(\sqrt{0} = 0\), the equation becomes:\[ x = \frac{-24}{18} \]Simplifying this fraction gives:\[ x = \frac{-4}{3} \]

Key Concepts

Distributive Property

The distributive property is a core concept in algebra that connects multiplication and addition (or subtraction). It states that multiplying a number by a sum is the same as multiplying the number separately by each addend, then adding the products. This property helps to break down and simplify complex expressions into more manageable parts.
In the given exercise, we start with the equation:

• \(3x(3x + 10) = 6x - 16\)

Applying the distributive property, we multiply each term inside the parentheses by \(3x\):

• \(3x \cdot 3x\) results in \(9x^2\)
• \(3x \cdot 10\) results in \(30x\)

Thus, the equation becomes \(9x^2 + 30x = 6x - 16\).
This transformation simplifies the equation, setting the stage for further manipulation and solution.

Simplifying Expressions

Simplifying expressions involves combining like terms and rearranging an equation to its simplest form. Once the distributive property is used, simplification is the next critical step that makes equations easier to solve and understand.
From our previous step using distribution, we have:

• Left side of the equation: \(9x^2 + 30x\)
• Equation: \(9x^2 + 30x = 6x - 16\)

By moving all terms to one side, we achieve:

• \(9x^2 + 30x - 6x + 16 = 0\)

We can combine like terms by subtracting and adding term coefficients:

• Combine \(30x\) and \(-6x\) to get \(24x\)

Thus, the simplified form of the equation is:

• \(9x^2 + 24x + 16 = 0\)

This form is particularly important for solving quadratics, as it prepares the equation for methods like factoring or applying the quadratic formula.

Quadratic Formula

The quadratic formula is a universal method used to find solutions to quadratic equations of the form \(ax^2 + bx + c = 0\). Quadratic equations often have two solutions, found using:

• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula provides the solution by accounting for possible roots through the "\(\pm\)" symbol, which indicates two possible values for \(x\).
In our solved equation:

• Coefficients: \(a = 9\), \(b = 24\), \(c = 16\)
• Substituting into the quadratic formula: \(x = \frac{-24 \pm \sqrt{24^2 - 4 \times 9 \times 16}}{2 \times 9}\)

This equation simplifies to:\(x = \frac{-24 \pm \sqrt{0}}{18}\). Here, \(\sqrt{0}\) simplifies the solution to a single root, \(x = -\frac{4}{3}\).
This approach covers all possible scenarios for quadratic equations and ensures accurate solutions, regardless of its complexity.