Problem 153 - Elementary Algebra

Question

Factor. \(11 x^{2}+34 x+3\)

Step-by-Step Solution

Answer

(11x + 1)(x + 3)

Step 1: Identify coefficients

Observe the quadratic equation and identify the coefficients. Here, the given quadratic equation is \[ 11x^2 + 34x + 3\] The coefficients are:\[ a = 11, \; b = 34, \; c = 3 \]

Step 2: Calculate the product of a and c

Multiply the coefficients a and c. \[ a \times c = 11 \times 3 = 33 \]

Step 3: Find two numbers that multiply to ac and add to b

Find two numbers that multiply to 33 and add to 34. These numbers are 33 and 1 because \[33 \times 1 = 33 \] \[33 + 1 = 34 \]

Step 4: Rewrite the middle term

Rewrite the middle term using the two numbers found. \[ 11x^2 + 34x + 3 = 11x^2 + 33x + x + 3 \]

Step 5: Factor by grouping

Group the terms in pairs and factor each group:\[ 11x^2 + 33x + x + 3 = 11x(x + 3) + 1(x + 3) \]

Step 6: Factor out the common binomial factor

Factor out the common binomial factor from the grouped terms: \[ 11x(x + 3) + 1(x + 3) = (11x + 1)(x + 3) \]

Key Concepts

quadratic equation

A quadratic equation is a polynomial equation of the second degree. This means the highest exponent of the variable (usually represented as x) is 2. A standard form of a quadratic equation looks like this: \[ax^2 + bx + c = 0\].

• Here, \(a, b,\) and \(c\) are coefficients.
• The term \(ax^2\) is known as the quadratic term.
• The term \(bx\) is the linear term.
• The term \(c\) is the constant term.

Understanding quadratic equations is crucial for solving problems in algebra, as they frequently appear in various math problems. In this exercise, the equation was \(11x^2 + 34x + 3\).

factoring by grouping

Factoring by grouping is an effective method used to factor quadratic equations when a simple factorization isn't immediately apparent. Here are the steps:

• First, identify the coefficients of the quadratic equation. For instance, in our given problem, \(a = 11, b = 34, c = 3\).
• Next, multiply the coefficients \(a\) and \(c\): \(11 \times 3 = 33\).
• Find two numbers that multiply to get the product of \(a\) and \(c\) and add up to the coefficient \(b\). In this example, we found that 33 and 1 work because \(33 \times 1 = 33\) and \(33 + 1 = 34\).
• Rewrite the middle term using these two numbers: \(11x^2 + 34x + 3 = 11x^2 + 33x + x + 3\).
• Group the terms: \(11x^2 + 33x + x + 3\) becomes \(11x(x + 3) + 1(x + 3)\).
• Factor out the common binomial factor: \((11x + 1)(x + 3)\).

Grouping simplifies the factorization process, especially when dealing with more complex quadratic equations.

coefficients

Coefficients are the numerical factors in the terms of a polynomial. In a quadratic equation in standard form \(ax^2 + bx + c\), the coefficients are crucial as they determine the specific characteristics of the equation. Here’s a breakdown:

• Coefficient \(a\): This is the coefficient of the quadratic term \(x^2\). In the example, \(a = 11\).
• Coefficient \(b\): This is the coefficient of the linear term \(x\). In the example, \(b = 34\).
• Coefficient \(c\): This is the constant term. In the example, \(c = 3\).

Understanding these coefficients and their appropriate manipulation is key to solving quadratic equations. It’s important to identify them correctly to apply methods like factoring by grouping efficiently.