Problem 27 - Basic Technical Mathematics with Calculus

Question

Solve the given quadratic equations by factoring. $$6 x^{2}=13 x-6$$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{2}{3} \) and \( x = \frac{3}{2} \).

Step 1: Rearrange Equation

The given equation is \( 6x^2 = 13x - 6 \). First, we need to move all terms to one side of the equation to set it to zero. So, we subtract \( 13x \) and add \( 6 \) to both sides resulting in: \( 6x^2 - 13x + 6 = 0 \).

Step 2: Factor the Quadratic Polynomial

Next, we need to factor the quadratic equation \( 6x^2 - 13x + 6 \) completely. We look for two numbers that multiply to \( 6 \times 6 = 36 \) and add up to \(-13\). These numbers are \(-9\) and \(-4\).

Step 3: Rewrite the Middle Term

Use the numbers found in Step 2 to split the \(-13x\) term. Rewrite the equation as: \( 6x^2 - 9x - 4x + 6 = 0 \).

Step 4: Factor by Grouping

Group the terms to factor by grouping. First group: \( (6x^2 - 9x) \), which factors to \( 3x(2x - 3) \). Second group: \( (-4x + 6) \), which factors to \( -2(2x - 3) \). The equation now reads: \( 3x(2x-3) - 2(2x-3) = 0 \).

Step 5: Factor Out the Common Binomial

Notice that \( (2x - 3) \) is common in both groups. Factor out \( (2x - 3) \): \( (3x - 2)(2x - 3) = 0 \).

Step 6: Solve for x

Set each factor equal to zero: \( 3x - 2 = 0 \) or \( 2x - 3 = 0 \). Solving for \( x \), we get: \( x = \frac{2}{3} \) or \( x = \frac{3}{2} \).

Key Concepts

Factoring

Factoring is a method used to solve quadratic equations by expressing a polynomial as a product of its factors. In simple terms, factoring involves breaking down a complex expression into simpler parts that, when multiplied together, give the original expression. This is especially helpful in solving quadratic equations of the form \( ax^2 + bx + c = 0 \).

To factor a quadratic equation:

• First, ensure the equation is set to zero by arranging all terms on one side.
• Identify two numbers that multiply to the product of the leading coefficient \( a \) and the constant term \( c \).
• These two numbers must also add up to the middle coefficient \( b \).

In the example given, \( 6x^2 - 13x + 6 = 0 \), we identified numbers \(-9\) and \(-4\) as the potential candidates.

By using factoring, we transformed the quadratic into \((3x - 2)(2x - 3) = 0\). Each factor represents a potential solution for the equation.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols. In solving equations, algebra is used to systematically rearrange expressions and isolate variables. One of its main goals is to find unknown values that satisfy given equations.

In our problem, algebraic techniques were used at several steps:

• First, rearranging the original equation \(6x^2 = 13x - 6\) to \(6x^2 - 13x + 6 = 0\) involved basic operations of adding and subtracting terms.
• During factoring, algebra was used to express the middle term as a sum of two terms, making it possible to simplify the expression through grouping.
• Finally, solving \(3x - 2 = 0\) and \(2x - 3 = 0\) involves isolating \(x\) by performing operations directly on the equation.

These techniques are fundamental to understanding and solving a wide range of mathematical problems, not just quadratic equations.

Polynomial

A polynomial is an expression that can have constants, variables, and exponents, combined using addition, subtraction, and multiplication. The highest exponent in a polynomial expression gives it a specific name, such as quadratic for degree two polynomials.

In the equation \(6x^2 - 13x + 6\), the expression is a quadratic polynomial because the highest degree is 2, due to the presence of the term \(6x^2\). Polynomials play a crucial role in algebra, as they are foundational elements in equations needing to be solved.

To solve the quadratic polynomial, recognizing the standard form \(ax^2 + bx + c\) is essential. Each coefficient and constant has a role:

• The leading coefficient \(a = 6\) affects the width and direction of the parabola represented by the equation.
• The middle term coefficient \(b = -13\) influences the shape and position of the parabola along the x-axis.
• The constant \(c = 6\) helps determine the y-intercept.

Understanding these characteristics allows for effective manipulation and transformation into a factorable form, simplifying the solution process.