Problem 21 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$3 x^{2}-13 x+4=0$$

Step-by-Step Solution

Answer

The solutions are \(x = \frac{1}{3}\) and \(x = 4\).

Step 1: Identify the Equation

We start with the quadratic equation: \(3x^2 - 13x + 4 = 0\). It is in the standard form \(ax^2 + bx + c = 0\) where \(a = 3\), \(b = -13\), and \(c = 4\).

Step 2: Find the Product and Sum

Calculate the product \(ac = 3 \times 4 = 12\) and the sum \(b = -13\). We need two numbers that multiply to 12 and add to -13.

Step 3: Determine the Pair of Numbers

The numbers that meet these conditions are \(-12\) and \(-1\) because \((-12) \times (-1) = 12\) and \((-12) + (-1) = -13\).

Step 4: Rewrite the Middle Term

Rewrite the middle term \(-13x\) as \(-12x - x\). The equation becomes \(3x^2 - 12x - x + 4 = 0\).

Step 5: Factor by Grouping

Group the terms: \((3x^2 - 12x) + (-x + 4) = 0\). Factor each group separately: \(3x(x - 4) - 1(x - 4) = 0\).

Step 6: Factor out the Common Binomial

Notice \((x - 4)\) is common in both groups: \((3x - 1)(x - 4) = 0\).

Step 7: Solve for x

Set each factor equal to zero: \(3x - 1 = 0\) and \(x - 4 = 0\). Solve both: \(x = \frac{1}{3}\) and \(x = 4\).

Key Concepts

Factoring

Factoring is a powerful algebraic tool that helps to simplify expressions and solve equations. This technique involves breaking down a complex expression into simpler components, called factors, that when multiplied together give the original expression.
For quadratic equations, factoring is often used to rewrite the equation in a form that can be easily solved. This is achieved by finding two binomials that multiply to give the original quadratic.
Here are some handy tips for factoring:

• Always identify a common factor in all terms if possible, and factor it out.
• Focus on the middle term and constant coefficient for quadratics, as you need two numbers whose product is the constant and sum is the middle coefficient.
• Check your work by expanding your factors to see if you get the original equation.

Factoring makes solving quadratic equations by further breaking them down into simpler linear equations.

Polynomial Equations

In algebra, polynomial equations are equations expressed in terms of polynomials. A polynomial is an expression consisting of variables, coefficients and the operations of addition, subtraction, and multiplication. Each term's variable may have non-negative integer exponents.
A quadratic equation is a specific type of polynomial equation where the highest exponent of the variable is 2. It's usually in the form:\[ ax^2 + bx + c = 0 \]where:

• \(a, b,\) and \(c\) are constants, and
• \(a eq 0\)

By dealing with these expressions, students develop problem-solving skills and a deeper understanding of mathematical relationships.
Understanding how to manipulate polynomial equations is crucial for solving more complex mathematical problems.

Solving Equations

Solving equations involves finding the values of the variables that make the equation true. For quadratic equations, this can be accomplished through various methods including factoring, using the quadratic formula, or completing the square.
Each method suits different types of quadratic equations, but factoring is particularly effective when an equation can be easily decomposed into binomial factors.

• Always start by writing the equation in standard form.
• Look for patterns or coefficients that are easy to manage.
• Set each factor equal to zero and solve for the variable to find your solutions.

To verify, substitute the solutions back into the original equation to ensure they satisfy it.
Through practice, recognizing which method to use and executing it becomes intuitive.

Algebraic Expressions

Algebraic expressions are combinations of constants, variables, and operations like addition, subtraction, multiplication, and division. These expressions form the foundation of equations and other mathematical statements.
Consider the role of each component:

• **Constants**: Fixed numbers in the expression (e.g., 3, 4).
• **Variables**: Symbols that represent numbers (e.g., \(x\)).
• **Coefficients**: Numbers multiplying the variables (e.g., \(3\) in \(3x\)).

Understanding these components makes it easier to manipulate expressions and develop problem-solving strategies.
In algebra, being comfortable with these expressions allows you to transform and simplify them to find solutions to equations.
Mastery over algebraic expressions is critical for success in more advanced mathematics.