Problem 19 - Algebra for College Students

Question

Solve each of the quadratic equations by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). If necessary, return to Chapter 3 and review the factoring techniques presented there. $$6 x^{2}-5 x-21=0$$

Step-by-Step Solution

Answer

Solutions are \(x = \frac{7}{3}\) and \(x = -\frac{3}{2}\).

Step 1: Identify Terms

The equation is given as \(6x^2 - 5x - 21 = 0\). Our goal is to express it in factored form as \((px + q)(rx + s) = 0\) by finding two binomials that multiply to produce the original quadratic.

Step 2: Factor the quadratic

First, find two numbers that multiply to \(a imes c = 6 imes (-21) = -126\) and add to \(-5\) (the coefficient of \(x\)). The numbers \(9\) and \(-14\) satisfy this, because \(9 imes (-14) = -126\) and \(9 + (-14) = -5\). Rewrite the middle term as \(9x - 14x\).

Step 3: Group and factor by grouping

Rewrite the quadratic as \(6x^2 + 9x - 14x - 21 = 0\). Group the terms: \((6x^2 + 9x) + (-14x - 21) = 0\). Factor out the common factors: \(3x(2x + 3) - 7(2x + 3) = 0\).

Step 4: Factor out the common binomial

Notice \((2x + 3)\) is a common factor in both groups. The equation can be factored to: \((3x - 7)(2x + 3) = 0\).

Step 5: Apply the zero-product property

Using the property \(ab = 0\) if and only if \(a = 0\) or \(b = 0\), solve each equation formed. Set each factor equal to zero: \(3x - 7 = 0\) and \(2x + 3 = 0\).

Step 6: Solve each equation for x

For \(3x - 7 = 0\), add 7 to both sides to get \(3x = 7\). Divide by 3 to find \(x = \frac{7}{3}\). For \(2x + 3 = 0\), subtract 3 from both sides to get \(2x = -3\). Divide by 2 to find \(x = -\frac{3}{2}\).

Key Concepts

Factoring

Factoring is a crucial step in solving quadratic equations, as it simplifies the equation and prepares it for solutions. A quadratic equation, such as \(6x^2 - 5x - 21 = 0\), can often be expressed as a product of two binomial expressions. The process of factoring involves breaking down the equation to find two numbers that fulfill two key conditions: they multiply to the product of the coefficient of \(x^2\) (here \(6\)) and the constant term (here \(-21\)), and they add up to the coefficient of \(x\), which is \(-5\).

• First, compute the product of \(a \times c\) where \(a\) is 6 and \(c\) is \(-21\). This gives us \(-126\).
• Find two integers that multiply to \(-126\) and add up to \(-5\). The numbers 9 and \(-14\) meet these conditions.
• Rewrite the middle term, \(-5x\), using the numbers 9 and \(-14\), so the equation becomes \(6x^2 + 9x - 14x - 21 = 0\).
• Next, use grouping to factor the equation: group the terms as \((6x^2 + 9x) + (-14x - 21)\) and factor each group.
• Factor out common factors to yield \(3x(2x + 3) - 7(2x + 3) = 0\).
• Notice how \((2x + 3)\) is a common factor, leading to a fully factored form of the quadratic: \((3x - 7)(2x + 3) = 0\).

Factoring, though sometimes challenging, is often a more straightforward method than using the quadratic formula and is particularly advantageous when integer solutions are possible.

Zero-Product Property

The zero-product property is a fundamental principle used after factoring a quadratic. It states that if the product of two factors is zero, then at least one of the factors must be zero. Applying this property is how we move from a factored quadratic equation to solving for its roots.

• Consider the factored form of our original equation: \((3x - 7)(2x + 3) = 0\).
• According to the zero-product property, this equation holds true if \((3x - 7) = 0\) or \((2x + 3) = 0\).
• The next step is to set each binomial equal to zero and solve for \(x\).
• For \(3x - 7 = 0\), add 7 to both sides to obtain \(3x = 7\), and then divide by 3, resulting in \(x = \frac{7}{3}\).
• For \(2x + 3 = 0\), subtract 3 from both sides to get \(2x = -3\), then divide by 2 to find \(x = -\frac{3}{2}\).

This property is extremely powerful in algebra, simplifying the process of solving equations by providing a direct route from factoring to solutions. It is crucial to remember that the zero-product property is applicable only when the product itself is set to zero.

Solving Quadratics

Solving quadratic equations is a fundamental skill in algebra that involves finding the values of \(x\) that make the equation equal to zero. These values are known as the roots or solutions of the equation. When an equation is set in the standard quadratic form, \(ax^2 + bx + c = 0\), several methods can be used to find these solutions, including factoring, completing the square, and using the quadratic formula.

• The most efficient method varies depending on the specific equation. In our example \(6x^2 - 5x - 21 = 0\), factoring was the preferred method.
• This approach involved transforming the equation into \((3x - 7)(2x + 3) = 0\) and then applying the zero-product property to find \(x = \frac{7}{3}\) and \(x = -\frac{3}{2}\).
• Factoring is often fastest when the coefficients of the equation allow for simple integer factor pairs.

However, not all quadratics are factorable using integers. For such equations, the quadratic formula \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\) can provide a reliable solution. Solving quadratics is about recognizing the most appropriate method for the equation at hand, which often involves practice and familiarity with different techniques.