Problem 132 - Algebra and Trigonometry Real Mathematics, Real People

Question

Factor the polynomial. $$16 x^{2}-54 x-7$$

Step-by-Step Solution

Answer

Upon finding the roots using the quadratic formula from step 2, the quadratic can be factored as in step 3. However, the roots are dependent on the solutions acquired from the quadratic formula as there isn't a straight forward method to factor the equation.

Step 1: Identify the quadratic

First, it can be seen that the equation given is a quadratic equation in the form of \(ax^{2} + bx + c\), where \(a = 16\), \(b = -54\), and \(c = -7\)

Step 2: Apply the quadratic formula

For any quadratic expression in the form \(ax^{2} + bx + c\), the roots can be found using the quadratic formula, which is \(-b \pm \sqrt{b^{2}-4ac} / 2a\). When we plug \(a = 16\), \(b = -54\), and \(c = -7\) into this equation, we get the roots.

Step 3: Factorize the quadratic expression

Knowing the roots of the corresponding quadratic equation, we can express the original quadratic expression in factored form. If we denote the roots by \(r_{1}\) and \(r_{2}\), then the factored version of the quadratic expression will be \(a(x-r_{1})(x-r_{2})\), to produce the equivalent form of the original given expression.

Key Concepts

Quadratic Equations

Quadratic equations are quite common in algebra, and they usually show up in the standard form: \( ax^2 + bx + c = 0 \). Why are they called 'quadratic'? Because the term "quadratic" comes from "quad," which means square. Quadratics involve the square of a variable, typically \( x^2 \). They hold significance because they can model real-world situations such as projectile motion or calculating areas.

• They always result in a parabolic graph which could be opening upwards or downwards.
• The solution or "roots" of a quadratic are the \( x \) values for which the expression equals zero.
• Quadratics can have either zero, one, or two real solutions.

Understanding the components \( a, b, \) and \( c \) helps in solving these equations. These coefficients guide us on how "wide" or "narrow" the parabola is, and how it is situated on the graph.

Quadratic Formula

The quadratic formula is a trustworthy tool for finding the roots of any quadratic equation. Derived from the process of completing the square, it is often a "go-to" method when factoring seems challenging. Given
\[ ax^2 + bx + c = 0 \]
The quadratic formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This formula allows you to calculate the roots by simply plugging in the values from the given quadratic equation.

• The term under the square root, \( b^2 - 4ac \), is known as the discriminant.
• The discriminant determines the nature of the roots:
  - If \( b^2 - 4ac > 0 \), there are two distinct real roots.
  - If \( b^2 - 4ac = 0 \), there is exactly one real root.
  - If \( b^2 - 4ac < 0 \), the roots are complex and not real numbers.

Using the quadratic formula might seem a little complicated at first, but it becomes straightforward with practice. It provides a reliable solution pathway when other methods are either difficult or infeasible.

Factoring Techniques

Factoring is a method used to express a polynomial as a product of its simpler polynomials, or "factors". This technique is especially useful because once a quadratic is factored, it's easy to find the roots by setting each factor equal to zero.
There are several techniques for factoring quadratics:

• **Trial and Error:** This intuitive method involves guessing and checking possible factors.
• **Using the Quadratic Formula:** Once the roots are known, the quadratic can be expressed in factored form \( a(x - r_1)(x - r_2) \), where \( r_1 \) and \( r_2 \) are the roots.
• **Difference of Squares:** This involves recognizing terms that can be expressed as \( (x-a)(x+a) \).
• **Factoring by Grouping:** Used when a quadratic expression does not have a leading coefficient of 1. Group terms to find common factors and simplify the expression.

Factoring requires understanding and practice, but it becomes a powerful technique especially when graphing or solving practical problems. Once mastered, it paves the path to simpler solution methods for solving quadratic equations.