Problem 40 - Algebra and Trigonometry

Question

31–76 ? Factor the expression completely. $$ 2 x^{2}+7 x-4 $$

Step-by-Step Solution

Answer

The expression is factored as \((2x - 1)(x + 4)\).

Step 1: Identify A, B, and C in the Quadratic

The given quadratic expression is in the form of \( ax^2 + bx + c \). In this expression: \[ a = 2, \, b = 7, \, c = -4 \] These coefficients will help us factor the quadratic.

Step 2: Calculate the Product 'ac'

Calculate the product of coefficients \( a \) and \( c \):\[ ac = 2 \times (-4) = -8 \] This product will be used to find two numbers that multiply to -8 and add to \( b = 7 \).

Step 3: Find Two Numbers that Multiply to 'ac' and Add to 'b'

Look for two numbers that multiply to \(-8\) and add up to \(7\). These numbers are \(8\) and \(-1\) because \[ 8 \times (-1) = -8 \] \[ 8 + (-1) = 7 \].

Step 4: Rewrite the Middle Term of the Quadratic Expression

Using the numbers found, rewrite the quadratic expression by splitting the middle term: \[ 2x^2 + 8x - x - 4 \].

Step 5: Factor by Grouping

Group the terms and factor separately: \[ (2x^2 + 8x) + (-x - 4) \] \[ 2x(x + 4) - 1(x + 4) \] Now, notice that \((x + 4)\) is a common factor.

Step 6: Factor the Common Term

Factor out the common term \((x + 4)\): \[ (2x - 1)(x + 4) \].This is the completely factored form of the expression.

Key Concepts

Quadratic Expressions

Quadratic expressions are a fundamental concept in algebra that typically take the form \( ax^2 + bx + c \). In this structure, \( a \), \( b \), and \( c \) are constants. Here, \( a \) cannot be zero because then the expression would not be quadratic. The "quadratic" portion comes from the \( x^2 \) term, which is the highest power of the variable in the expression.

Understanding quadratic expressions is crucial because they appear in various mathematical problems, including physics, engineering, and finance. They often model parabolic relationships where the direction and width of the parabola are determined by the coefficient \( a \).

For the given expression \( 2x^2 + 7x - 4 \), identifying the coefficients \( a = 2 \), \( b = 7 \), and \( c = -4 \) is the first step in many quadratic problem-solving processes, such as factoring.

Factoring by Grouping

Factoring by grouping is a powerful technique used especially for factoring quadratics when direct methods are not feasible. This method involves splitting the middle term of the quadratic expression and rearranging terms so that they can be grouped into pairs that share a common factor.

Let's consider the quadratic \( 2x^2 + 7x - 4 \). After rewriting the middle term using numbers that multiply to \( ac \) and add to \( b \), you split it into two terms, resulting in \( 2x^2 + 8x - x - 4 \).

Next, you group these terms into two pairs:

• \( (2x^2 + 8x) \)
• \( (-x - 4) \)

By factoring each group separately, you pull out common factors:

• From \( 2x^2 + 8x \), extract \( 2x(x + 4) \)
• From \( -x - 4 \), extract \( -1(x + 4) \)

This results in recognizing a common binomial factor \((x + 4)\), which can then be factored out, leading to the fully factored expression \((2x - 1)(x + 4)\).

Polynomial Factoring

Polynomial factoring is a critical skill in algebra that simplifies expressions by breaking them into a product of polynomials. In the case of quadratics, factoring transforms expressions into the form \( (mx + n)(px + q) \), allowing for easier solving of equations or simplification of expressions.

The polynomial, \( 2x^2 + 7x - 4 \), required several steps of factoring by grouping to reach its simplest form. This simplifies the quadratic into two linear factors: \((2x - 1)(x + 4)\). Factoring is essential because it sometimes allows quadratic equations to be solved by setting each factor equal to zero, leading to solutions through the Zero Product Property.

Remember, factoring quadratics not only aids in solving equations but also in gaining insights into the behavior of functions, such as intercepts and roots of the corresponding quadratic function.