Problem 111 - Algebra 2

Question

Factor each expression. $$ 5 x^{2}+13 x-6 $$

Step-by-Step Solution

Answer

The factorized expression is \((x + 3)(5x - 2)\)

Step 1: Identification of factors

The first step is to identify two numbers that multiply to -30 (5*-6) and Add to 13. The numbers that fit these conditions are 15 and -2, because 15*-2 = -30 and 15 + (-2) = 13.

Step 2: Rewrite the middle term

The next step is to rewrite the middle term in the equation using these numbers. Instead of writing 13x, write it as 15x - 2x. The equation then becomes \(5x^2 + 15x - 2x - 6\).

Step 3: Factor by grouping

Now, perform factorization by grouping. Group the first two terms together and the last two terms together. This gives \((5x^2 + 15x) - (2x + 6)\). From the first group, take out the common factor, which is \(5x\), and from the second, take out -2. We have \(5x(x + 3) - 2(x + 3)\).

Step 4: Common Factor

Now, notice that \((x+3)\) is a common factor. So, take out \((x + 3)\) as common, which gives \((x + 3)(5x - 2)\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation characterized by a degree of 2. It typically takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. In the above exercise, we are dealing with the expression

• \( 5x^2 + 13x - 6 \)

which is not set to zero, but it follows the quadratic form with \( a = 5 \), \( b = 13 \), and \( c = -6 \). Quadratic equations can have up to two real solutions, which are found when the expression equals zero.

Solving a quadratic equation can be done using various methods such as factoring, completing the square, or using the quadratic formula. Factoring is often the simplest method if the polynomial can be easily expressed as the product of two binomials. Being able to rewrite a quadratic expression in factored form can reveal its roots, providing valuable insight into the values of \( x \) that make the equation true.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain numbers, variables, and operations. In contrast to equations, they do not have an equals sign.

Algebraic expressions like \( 5x^2 + 13x - 6 \) form the backbone of algebra and are integral to understanding polynomial functions. Such expressions demand manipulation and simplification through operations such as addition, subtraction, multiplication, and factoring.

• Variables represent unknown values.
• Coefficients are numbers multiplying the variables.
• Constants are standalone numerical values.

Understanding the parts of an algebraic expression is crucial as it aids in performing operations like factoring. Through methods akin to what we saw in the original solution, these expressions can be restructured to simplify the solving process, which is essential in numerous areas of mathematics and applied sciences.

Polynomial Factoring

Polynomial factoring is a key algebraic skill used to express a polynomial as a product of its factors. It simplifies expressions and equations, allowing us to more easily find solutions or roots.

To factor a quadratic expression such as \( 5x^2 + 13x - 6 \), we use the process demonstrated in the step-by-step solution:

• Identification of Factors: Determine two numbers that multiply to the product of the leading coefficient and the constant term, and add up to the middle coefficient.
• Rewrite the Middle Term: Use the identified numbers to split the middle term.
• Group and Factor: Group the terms and factor out the common factors.
• Common Factor: Identify any common binomial, allowing us to consolidate the expression into the factored form.

This method brings the expression to a form such as \((x + 3)(5x - 2)\), revealing potential solutions for \( x \). In quadratic relationships, these solutions directly relate to the zeros or roots, which indicate where the polynomial expression equals zero. Factoring is a powerful algebraic tool used across various domains to break down and simplify expressions.