Problem 27 - Elementary Algebra

Question

Factor completely. $$ -7 x 2+19 x+6 $$

Step-by-Step Solution

Answer

The expression factors to \((-x + 3)(7x + 2)\).

Step 1: Write down the expression

We have the expression \[-7x^2 + 19x + 6\] which we need to factor.

Step 2: Identify coefficients

The quadratic expression is in the form \(ax^2 + bx + c\). Here, \(a = -7\), \(b = 19\), and \(c = 6\).

Step 3: Multiply \(a\) and \(c\)

Calculate the product of \(a\) and \(c\): \(-7 \times 6 = -42\).

Step 4: Find two numbers that multiply to \(-42\) and add to \(19\)

We need two numbers that multiply to \(-42\) and add up to \(19\). These numbers are \(-2\) and \(21\), as \(-2 \times 21 = -42\) and \(-2 + 21 = 19\).

Step 5: Rewrite the middle term using these numbers

Rewrite the expression by splitting the middle term:\[-7x^2 - 2x + 21x + 6\].

Step 6: Factor by grouping

Group the terms in pairs and factor each pair:1. From \(-7x^2 - 2x\), factor out \(-x\):\(-x(7x + 2)\).2. From \(21x + 6\), factor out \(3\):\(3(7x + 2)\).So the expression becomes:\((-x + 3)(7x + 2)\).

Step 7: Verify the factorization

Expand the factors to verify the solution:\((-x + 3)(7x + 2) = -7x^2 + 2x + 21x + 6 = -7x^2 + 19x + 6\). This confirms our factorization is correct.

Key Concepts

Quadratic Expression

A quadratic expression is a polynomial that includes a variable raised to the power of two as its highest degree. It usually fits the general form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. Quadratic expressions are fundamental in both algebra and geometry, representing parabolas when graphed. The key terms in these expressions are:

• Quadratic Term: This is the \(ax^2\) part, which is the term with the variable squared.
• Linear Term: The \(bx\) term, which is the term with the variable raised to the power of one.
• Constant Term: The \(c\) term, which is a plain number without variables.

Recognizing these components helps in identifying and manipulating quadratic equations. For example, in the expression \(-7x^2 + 19x + 6\), \(a = -7\), \(b = 19\), and \(c = 6\). Understanding each component lays the groundwork for solving quadratic equations through various methods, including factoring.

Factoring by Grouping

Factoring by grouping is a method often used with quadratic expressions when simple factoring isn't straightforward. This technique works by rearranging and splitting the quadratic into groups that can each be factored separately. Here’s how it generally works:

• First, find two numbers that multiply to the product of \(a\) and \(c\), and add up to \(b\).
• Rewrite the expression by breaking up the linear term \(bx\) using these two numbers.
• Next, group the terms into pairs and factor out a common factor from each pair.
• Finally, factor out the remaining common factor from the two newly formed terms.

For the expression \(-7x^2 + 19x + 6\), factoring by grouping involves rewriting the middle term into two terms, resulting in \(-7x^2 - 2x + 21x + 6\). This allows you to group and factor separately: \(-x(7x + 2) + 3(7x + 2)\). The common factor \((7x + 2)\) is then factored out, yielding the solution \((-x + 3)(7x + 2)\). Factoring by grouping can simplify complex expressions and reveal their factors clearly.

Quadratic Coefficients

In quadratic expressions, the coefficients \(a\), \(b\), and \(c\) play a crucial role in solving these polynomials. The coefficient \(a\) affects the direction and width of the parabola when graphed. If \(a\) is negative, like in \(-7x^2\), the parabola opens downwards. For factoring, \(a\) influences both the product used to find pairing numbers and how terms are split. The coefficient \(b\) is critical in determining the middle term's effect and how solutions are derived in factorization. The value of \(c\) is the intercept, impacting where the graph crosses the y-axis.
In the given quadratic \(-7x^2 + 19x + 6\), unraveling these coefficients helps in applying the right factoring technique to solve it completely. Calculating \(a \times c\), which is essential in the factoring by grouping process, yields \(-42\). Finding two numbers that multiply to this value and sum to \(b = 19\) is a vital step. Understanding coefficients allows one to perceive how each term of the quadratic expression influences the overall expression and shapes its solution.