Problem 13 - Algebra and Trigonometry

Question

Solve the equation by factoring. $$ 6 x^{2}+5 x=4 $$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{2}\) and \(x = -\frac{4}{3}\).

Step 1: Rearrange the Equation

Start by moving all terms to one side of the equation to set it to zero:\[ 6x^2 + 5x - 4 = 0 \]

Step 2: Apply the Factoring Method

To factor the quadratic equation, we need to find two numbers that multiply to \(-24\) (the product of \(6\) and \(-4\)) and add to \(5\). These numbers are \(8\) and \(-3\).

Step 3: Rewrite and Regroup the Equation

Replace \(5x\) in the equation with \(8x - 3x\) and regroup:\[ 6x^2 + 8x - 3x - 4 = 0 \]\[ (6x^2 + 8x) + (-3x - 4) = 0 \]

Step 4: Factor by Grouping

Factor out the greatest common factors from each group:\[ 2x(3x + 4) - 1(3x + 4) = 0 \]

Step 5: Factor Out the Common Binomial

Since \((3x + 4)\) appears in both groups, factor it out:\[ (2x - 1)(3x + 4) = 0 \]

Step 6: Solve Each Factor for Zero

Set each factor equal to zero and solve for \(x\):1. \(2x - 1 = 0\) - \(2x = 1\) - \(x = \frac{1}{2}\)2. \(3x + 4 = 0\) - \(3x = -4\) - \(x = -\frac{4}{3}\)

Key Concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). In our example, the equation \( 6x^2 + 5x - 4 = 0 \) is quadratic because it includes the term \( x^2 \), the highest power of \( x \). This form is crucial because it tells us that the graph of this equation is a parabola. Magnitudes of \( a \), \( b \), and \( c \) will determine properties like its direction (opening up or down) and the width. Dealing with a quadratic equation often involves factoring it into simpler binomials, which is what we will focus next. Understanding this structure is the key to solving it.

Factoring by Grouping

Factoring by grouping is a handy method when a quadratic equation cannot be easily factored by simple factoring techniques. Here's how it works:

• First, rearrange the equation if necessary to set it to zero.
• Next, the equation is split into two pairs of terms. For the example \( 6x^2 + 5x - 4 = 0 \), our split is \( 6x^2 + 8x \) and \( -3x - 4 \).
• The goal is to factor out the greatest common factor (GCF) from each pair separately. From \( 6x^2 + 8x \), the GCF is \( 2x \); from \( -3x - 4 \), it is \(-1\).
• Then, because each factor group should contain a common binomial factor (in this case \((3x + 4)\)), you can factor out this common pair to get \( (2x - 1)(3x + 4) = 0 \).

This method elegantly transforms the complex quadratic equation into a product of factors that can be individually set to zero.

Solving Equations

Once the quadratic equation is factored successfully, solving the equation becomes straightforward. Each factor is individually set equal to zero. In our example, the factors are \((2x - 1)\) and \((3x + 4)\), resulting in two simple linear equations:

• \( 2x - 1 = 0 \)
• \( 3x + 4 = 0 \)

Solving these grants us the roots of the quadratic equation:

• From \( 2x - 1 = 0 \) we add \(1\) to both sides, giving \( 2x = 1 \). Dividing by \(2\), we find \( x = \frac{1}{2} \).
• From \( 3x + 4 = 0 \), subtracting \(4\) results in \( 3x = -4 \). Dividing by \(3\), we get \( x = -\frac{4}{3} \).

These two solutions \( x = \frac{1}{2} \) and \( x = -\frac{4}{3} \) are where the parabola intersects the x-axis. This straightforward process is how we solve quadratic equations by factoring.