Problem 78 - College Algebra

Question

Exercises \(78-80\) will help you prepare for the material covered in the next section. Factor: $$ 2 x^{2}+7 x-4 $$

Step-by-Step Solution

Answer

The factorised form of the expression \(2x^2 + 7x - 4\) is \((2x - 1)(x + 4)\).

Step 1: Identifying the coefficients

First, identify the coefficients \(a\), \(b\), and \(c\) from the quadratic equation \(2x^2 + 7x - 4\). Here, \(a = 2\), \(b = 7\), and \(c = -4\).

Step 2: Finding the right pair of numbers

Now, we need to find two numbers that multiply to give the product \(ac = (2)(-4) = -8\), and that also add up to \(b = 7\). Two such numbers happen to be \(-1\) and \(8\), because \(-1 * 8 = -8\) and \(-1 + 8 = 7\).

Step 3: Rewrite the equation

The original equation can be rewritten by splitting the middle term: \(2x^2 + 8x - x - 4\).

Step 4: Factoring by grouping

Now, we use factoring by grouping. We group the first two terms and the last two terms separately: \(2x(x + 4) - 1(x + 4)\).

Step 5: Distributive property

We now factor by using the distributive property and we will get our final factorised equation: \((2x - 1)(x + 4)\).

Key Concepts

Quadratic Equations

Quadratic equations are mathematical expressions that take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown variable. These equations are "quadratic" because the highest exponent of \(x\) is 2. Quadratics are fundamental in algebra and appear in many real-world problems, from physics to finance.

To solve a quadratic equation, we often need to find the values of \(x\) that make the equation true. These solutions can be found using various methods like factoring, using the quadratic formula, or completing the square. For instance, factoring involves rewriting the quadratic as a product of two binomials. This approach reveals the roots or solutions of the equation.

Factoring by Grouping

Factoring by grouping is a technique used to break down a polynomial into simpler components or factors. This method works well for quadratics where direct factoring is not readily apparent.

• Start by identifying a way to split the middle term in such a way that grouping becomes possible.
• Divide the polynomial into two separate grouping sections.
• Factor out the common factor from each group.

In the case of \(2x^2 + 7x - 4\), we split the middle term to create two smaller expressions: \(2x^2 + 8x\) and \(-x - 4\). Each group is then factored independently to reveal common factors, simplifying the equation into a product of two binomials.

Distributive Property

The distributive property is a key algebraic rule that allows us to multiply a single term by terms inside parentheses. It states that \(a(b + c) = ab + ac\). This property helps simplify expressions and solve equations.

When applying the distributive property to factor a quadratic, we look for common factors in grouped terms and apply the property to factor them out. For \(2x(x + 4) - 1(x + 4)\), we notice \((x + 4)\) is common in both terms, allowing us to factor it out:

\((2x - 1)(x + 4)\)

This step confirms the successful application of the distributive property, ultimately simplifying the quadratic into product form.

Coefficients in Algebra

Coefficients are the numerical factors of the terms in an algebraic expression. In a quadratic equation like \(ax^2 + bx + c\), the coefficients are \(a\), \(b\), and \(c\). Understanding coefficients is crucial when working with algebraic expressions.

• Coefficient \(a\) determines the "width" and the direction of the parabola represented by the quadratic equation.
• Coefficient \(b\) influences the position and orientation of the parabola.
• Coefficient \(c\) represents the y-intercept when the quadratic is graphed.

Analyzing the coefficients helps us understand and factor the expression. For example, in \(2x^2 + 7x - 4\), recognizing \(a = 2\), \(b = 7\), and \(c = -4\) directs how we manipulate the quadratic during the factoring process.