Question

Factor: \(2 x^{2}-x-3\)

Short answer

The factored form is \((2x - 3)(x + 1)\).

Step-by-step solution

- Identify coefficients

Identify the coefficients of the quadratic equation. For the equation \(2x^2 - x - 3\), the coefficients are: \(a = 2\), \(b = -1\), and \(c = -3\).

- Find the product of a and c

Multiply the leading coefficient \(a\) and the constant term \(c\): \(2 \times -3 = -6\)

- Determine two numbers that multiply to ac and add to b

Find two numbers that multiply to \(-6\) (the result from Step 2) and add to \(-1\) (the value of \(b\)). These numbers are \(-3\) and \(2\), since \((-3) \times 2 = -6\) and \((-3) + 2 = -1\).

- Rewrite the middle term using these two numbers

Rewrite the quadratic equation by splitting the middle term \(-x\) into two terms using the numbers found in Step 3: \(2x^2 - 3x + 2x - 3\).

- Factor by grouping

Group the terms to factor by grouping: \((2x^2 - 3x) + (2x - 3)\). Factor out the greatest common factor (GCF) from each group: \(x(2x - 3) + 1(2x - 3)\).

- Factor out the common binomial factor

Factor out the common binomial factor \(2x - 3\): \((2x - 3)(x + 1)\).

Key concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that have the standard form: \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients where \(a eq 0\). The term \(ax^2\) is called the quadratic term, \(bx\) the linear term, and \(c\) the constant term. Quadratic equations often have two solutions, which can be found using various methods like the quadratic formula, completing the square, or factoring.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials by grouping terms that have common factors. This technique is especially useful for quadratic equations when the middle term can be split in such a way that groups can be formed with common factors. Let's break down how this is done with the quadratic equation \(2x^2 - x - 3\):

• First, identify two numbers that multiply to the product of the first and last coefficients (\(a\cdot c\)), and add up to the middle coefficient \(b\).
• In our example, those numbers are \(-3\) and \(2\) because \((-3) \times 2 = -6\) and \((-3) + 2 = -1\).
• Next, rewrite the equation by splitting the middle term using these two numbers: \(2x^2 - 3x + 2x - 3\).

Doing these steps correctly allows you to create groups and factor out the common factors effectively.

Coefficients

Coefficients are the numerical factors in the terms of a polynomial. In the quadratic equation \(2x^2 - x - 3\), the coefficients are:

• \(a = 2\) which is the coefficient of the quadratic term \(2x^2\)
• \(b = -1\) which is the coefficient of the linear term \(-x\)
• \(c = -3\) which is the constant term.

Understanding the role of coefficients is crucial because they determine the behavior and shape of the quadratic graph. For instance, the leading coefficient \(a\) influences the width and direction of the parabola.