Problem 12 - College Algebra

Question

Solve the quadratic equation by factoring. $$ 8 x^{2}+6 x-9=0 $$

Step-by-Step Solution

Answer

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

Step 1: Set Up the Equation

Let's start by making sure that the quadratic equation is in standard form, which is already the case here: \[ 8x^2 + 6x - 9 = 0 \] This equation is ready to be solved by factoring.

Step 2: Find Two Numbers

We need two numbers that multiply to \(8 \times -9 = -72\) and add to \(6\), the coefficient of \(x\). These numbers are \(12\) and \(-6\) because \(12 \times -6 = -72\) and \(12 + (-6) = 6\).

Step 3: Rewrite the Middle Term

Rewrite the middle term \(6x\) using the two numbers found in Step 2: \[ 8x^2 + 12x - 6x - 9 = 0 \].

Step 4: Factor by Grouping

Group the terms to factor them separately: \[ (8x^2 + 12x) + (-6x - 9) = 0 \].Factor out the greatest common factor from each group:\[ 4x(2x + 3) - 3(2x + 3) = 0 \].

Step 5: Factor Out the Common Binomial

Notice that \(2x + 3\) is a common binomial factor:\[ (4x - 3)(2x + 3) = 0 \].

Step 6: Solve for the Roots

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

Key Concepts

Factoring

Factoring is a method used in algebra to break down more complex expressions into simpler parts that, when multiplied together, give the original expression. In the context of quadratic equations, factoring involves finding two binomials that multiply to give the quadratic expression. This process is helpful because it can turn a difficult equation into simpler expressions, making it easier to solve for the variable, typically denoted as \( x \). Applying factoring to quadratic equations requires understanding how to decompose the quadratic into binomial factors, by looking for numbers that multiply to the product of the leading coefficient and the constant term and also add to the middle term's coefficient. This method is practical, especially when the quadratic equation does not lend itself easily to other solving methods.

Standard Form

The standard form of a quadratic equation is essential for identifying the coefficients and terms needed to solve the equation. It is generally expressed as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable.

• \( a \) is the coefficient of the squared term \( x^2 \).
• \( b \) is the coefficient of the linear term \( x \).
• \( c \) is the constant term.

In our specific equation \( 8x^2 + 6x - 9 = 0 \), \( a = 8 \), \( b = 6 \), and \( c = -9 \). Writing the equation in standard form is the first step in analyzing and solving it, giving us a clear pathway to factoring or using other methods like completing the square or the quadratic formula.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). These roots represent the points at which the quadratic graph, a parabola, crosses the x-axis. Finding the roots is equivalent to solving the quadratic equation and is often achieved through factoring, completing the square, or the quadratic formula. When a quadratic is factored into the form \((mx + n)(px + q) = 0\), the roots are found by setting each factor to zero:

• \( mx + n = 0 \)
• \( px + q = 0 \)

Thus, solving these linear equations gives the values of \( x \), which are the roots. In our example, the calculation yields \( x = \frac{3}{4} \) and \( x = -\frac{3}{2} \), meaning those are the points where the parabola intersects the x-axis.

Factoring by Grouping

Factoring by grouping is a versatile technique useful when dealing with quadratic expressions that are not readily factorable by simple methods. It involves breaking up the middle term into two terms that can be grouped with other terms to identify common factors.The process is as follows:1. Write the equation so that the like terms can be grouped together, maintaining equality.2. Within each group, factor out the greatest common factor (GCF).3. You'll often find a common binomial factor in the resulting expression.4. Finally, factor out the common binomial, simplifying it to a pair of binomial factors.Using the example equation \( 8x^2 + 6x - 9 = 0 \), these steps were used to achieve \((4x - 3)(2x + 3) = 0\). Factoring by grouping can sometimes reveal solutions that aren't immediately evident, providing a methodical approach to solving more complex quadratics.