Problem 15 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The solutions to the quadratic equation \(6x^2 - x - 2 = 0\) are \(x = 2/3\) and \(x = -1/2\).

Step 1: Factoring Quadratic Equation

Factor the quadratic equation \(6x^{2} - x - 2\) into two binomial expressions. This can be done by finding two numbers that multiply to \(6 * -2 = -12\), the product of 'a' and 'c' in the standard form, and add up to -1, the coefficient of 'x'. Those numbers are -4 and 3. Now rewrite the middle term as \(-4x + 3x\). Now the equation will be \(6x^2 - 4x + 3x - 2 = 0\).

Step 2: Factor by Grouping

Now that the quadratic is in four terms, we can factor by grouping. The first two terms can be factored by \(2x\), which gives out \(2x(3x - 2)\). The last two terms have a common factor of 1, which gives out \(1(3x - 2) = 0\)

Step 3: Common Factor

Notice that \((3x - 2)\) is a common factor. So, it becomes \((3x - 2)(2x + 1) = 0\). This is the factored form of the equation.

Step 4: Solve for x

Set each factor equal to zero and solve for x. Therefore, \(3x - 2 = 0\) and \((2x + 1) = 0\). Solving these gives out \(x = 2/3\) and \(x = -1/2\).

Key Concepts

Factoring Quadratic Equations

Factoring quadratic equations is a method used to solve quadratic equations, which are equations of the second degree, typically in the form of \(ax^2 + bx + c = 0\). The goal of factoring is to break down the equation into simpler factors that, when multiplied together, give the original quadratic equation. To factor a quadratic equation, one seeks two binomial expressions whose product is the original quadratic. This often involves finding two numbers that multiply to \(ac\) (the product of the 'a' and 'c' coefficients in the standard form) and sum to 'b' (the linear coefficient).

Example:

In the equation \(6x^2 - x - 2 = 0\), we look for two numbers that multiply to \(6 \times (-2) = -12\) and add up to \(-1\), the coefficient of 'x'. The numbers that meet these criteria are -4 and 3. Rewriting the quadratic with these numbers yields \(6x^2 - 4x + 3x - 2 = 0\). Factoring is not complete until a binomial form is revealed and no further simplification is possible.

Understanding this concept is vital, as it is a fundamental skill in algebra that enables the solving of many types of equations. In addition, it often lays the groundwork for more advanced topics in mathematics.

Factor by Grouping

Factor by grouping is a technique used in algebra to factor certain polynomials that have four or more terms. The method involves gathering the terms into two or more pairs and factoring out the greatest common factor from each pair. The key to successful factor by grouping is to rearrange the terms if necessary so that a common binomial factor emerges from the grouped terms.

Applying the Method:

Let's take the earlier equation \(6x^2 - 4x + 3x - 2 = 0\). We group the terms to reveal a common factor within each group: \(2x\) from the first group and \(1\) (or nothing) from the second group, leading us to \(2x(3x - 2) + 1(3x - 2) = 0\). Now, since \((3x - 2)\) is a common factor, we can factor it out, resulting in \((3x - 2)(2x + 1) = 0\), the factored form of the quadratic equation. Therefore, grouping can significantly simplify factoring when dealing with more complex equations.

Zero Product Property

The zero product property is a fundamental principle in algebra which states that if the product of two factors is zero, then at least one of the factors must be zero. This property is indispensable when solving quadratic equations by factoring because, after the equation is factored into binomials, we can set each factor equal to zero to find the solutions.

Putting It to Use:

With the factored equation \((3x - 2)(2x + 1) = 0\), we apply the zero product property. We set each factor equal to zero: \(3x - 2 = 0\) and \(2x + 1 = 0\), which gives us the solutions for 'x' after solving these simple linear equations. In this case, the solutions are \(x = 2/3\) and \(x = -1/2\). The zero product property ensures that we can identify all possible solutions to the equation, making it a powerful tool in the algebraic toolbox. Its application guarantees that no solution is missed, thus exhaustively solving the quadratic equation.