Problem 176 - Elementary Algebra

Question

Factor. \(6 p^{2}+p-22\)

Step-by-Step Solution

Answer

(6p + 11)(p - 2)

Step 1: Arrange the quadratic equation

Write the given quadratic equation in standard form, which is already arranged as: \[6p^2 + p - 22\]

Step 2: Identify coefficients

Identify the coefficients in the quadratic equation. Here, the coefficients are: \[a = 6\], \[b = 1\], \[c = -22\]

Step 3: Multiply 'a' and 'c'

Multiply the coefficient of \(p^2\) (which is \(a\)) with the constant term (which is \(c\)). \[a \times c = 6 \times (-22) = -132\]

Step 4: Find factor pair

Find a pair of factors of \(-132\) that add up to the coefficient of \(p\) (which is 1). The factor pair that satisfies this is: -11 and 12, because: \(-11 + 12 = 1\).

Step 5: Split the middle term

Rewrite the middle term (\(p\)) using the factor pair found:\[6p^2 - 11p + 12p - 22\]

Step 6: Factor by grouping

Group the terms in pairs and then factor out the common factor from each pair:\[6p(p - 2) + 11(p - 2)\]

Step 7: Factor out the common binomial

Factor the expression by taking out the common binomial \((p - 2)\):\[(6p + 11)(p - 2)\]

Key Concepts

Quadratic Equations

Quadratic equations are a key concept in algebra. A quadratic equation is any equation that can be written in the standard form: \({a x^2 + b x + c = 0}\).
Here,

• a: is the coefficient of the square term.
• b: is the coefficient of the linear term.
• c: is the constant term.

Understanding this form helps in rearranging and solving quadratic equations. Quadratics are common in many real-life applications, such as calculating area or understanding projectile motion. It’s crucial to get comfortable with recognizing and manipulating these forms.

Factoring

Factoring is a method used in algebra to simplify expressions and solve equations. When factoring a quadratic equation like \(6p^2 + p - 22\), our goal is to rewrite it as a product of simpler expressions. Here’s a simplified guide to factoring:

• Step 1: Identify the quadratic, linear, and constant terms.
• Step 2: Find two numbers that multiply to give \(a \times c\) and add to give b.
• Step 3: Split the 'b' term using these numbers.
• Step 4: Group the terms and factor each group.
• Step 5: Factor out the common binomial.

This makes the expression easier to solve or further manipulate.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In our exercise, the expression \(6p^2 + p - 22\) is an algebraic expression. The goal is to simplify it into a more manageable form. Here are some key points:

• Terms: These are parts of the expression separated by addition or subtraction. In \(6p^2 + p - 22\), there are three terms.
• Coefficients: Numbers that multiply the variables. For example, in \(6p^2\), 6 is the coefficient.
• Constant: A term without a variable. Here, it’s \(-22\).

Understanding these components helps in manipulating and solving algebraic expressions.

Coefficient Identification

Identifying coefficients is fundamental when working with algebraic expressions. For the quadratic equation \(6p^2 + p - 22\), we need to identify the coefficients to proceed with factoring.

• a: This is the coefficient of \(p^2\). In our case, \(a = 6\).
• b: This is the coefficient of \(p\). For \(6p^2 + p - 22\), \(b = 1\).
• c: This is the constant term, which is \(c = -22\).

Once we identify a, b, and c, we can use them to apply factoring methods. This involves finding two numbers that multiply to \(a \times c\) and add to b.