Problem 34 - Beginning and Intermediate Algebra

Question

Factor by trial and error. $$7 a^{2}-17 a+6$$

Step-by-Step Solution

Answer

The factored form of the given expression, \(7a^2 - 17a + 6\), is \((a+3)(7a-2)\), found using the trial and error method.

Step 1: Write the given quadratic expression

The given quadratic expression is: \(7a^2 - 17a + 6\)

Step 2: Factors of the first term and third term

We'll now find the factors of the leading term \(7a^2\) and the constant term \(6\). First term, \(7a^2\): - \(7а^2 = 1a * 7а\) Third term, 6: - \(6 = 1 * 6\) - \(6 = 2 * 3\)

Step 3: Trial and error

Now we'll try different combinations of factors of the first and third term to check for which combinations the multiplication results in the given expression. Let's try combining the factors as follows: \(1a * (7a + 2)\) \(3 * (1a + 6)\) When we perform the multiplication, we get: \(1a * 7a \Rightarrow 7а^2\) \(1a * 2 \Rightarrow 2а\) \(3 * 1a \Rightarrow 3а\) \(3 * 6 \Rightarrow 18\) Now, let's add the products: \(7а^2 + 2а + 3а + 18 \Rightarrow 7а^2 - 17а + 6\) So, this combination works!

Step 4: Write the factored form

The factored form of the given expression is: \((a+3)(7а-2)\)

Key Concepts

Trial and Error Method

The trial and error method is a practical approach to solving problems by testing various possibilities until a successful solution is found. In mathematics, especially when factoring quadratic expressions, this method involves combining different possible factors to determine which combination results in the original expression.
This approach is particularly useful when the expression doesn't readily reveal straightforward factor pairs. Although it might seem like a time-consuming process, trial and error enhances understanding and fosters problem-solving skills. It requires patience and persistence, but it often leads to the correct solution even for complex problems.

Factorization

Factorization is the process of breaking down an expression into multiplying factors that are simpler or more manageable. For quadratic expressions, factorization involves expressing the quadratic in the form \((ax + b)(cx + d)\), where these factors multiply to give the original quadratic expression.

• It simplifies the expression by rewriting it as a product of simpler expressions.
• This method makes solving equations easier since we can apply the zero-product property to find the solutions for "x" where each factor equals zero.

Factorization is crucial not only for solving quadratic equations but also for simplifying expressions in algebra.

Quadratic Expressions

Quadratic expressions are mathematical expressions of the form \(ax^2 + bx + c\), where "a", "b", and "c" are constants and "x" represents the variable. These expressions are central in algebra and help describe parabolic graphs.

• They include terms with the second power of a variable (squared terms).
• They can be factored, expanded, or solved for specific values of "x".

Quadratic expressions arise in various real-world problems, including physics, engineering, and finance, making them essential for students to understand.

Algebraic Expression

An algebraic expression is a combination of numbers, variables, and operators such as addition and multiplication. These expressions form the building blocks of algebra and can include terms, constants, coefficients, and variables.

• Variables in algebraic expressions represent unknown values and are denoted by symbols like "x" or "a".
• Constants are fixed numerical values that do not change.
• Coefficients are the numerical factors multiplied by the variable terms in the expression.

Understanding algebraic expressions is vital because they express mathematical relationships in a form that can be manipulated and used to solve equations and problem statements.