Factoring and Solving Quadratic Equations: A Step-by-Step Guide

In mathematics, factoring is a key skill in solving various types of equations, particularly quadratic equations. This article will guide you through the process of factoring and solving a specific quadratic equation, showing you how to simplify and find the solutions step-by-step.

Introduction to Factoring and Quadratic Equations

Quadratic equations are equations of the form ax^2 bx c 0, where a, b, and c are constants and a eq 0. The solutions of these equations can be found by various methods including factoring, the quadratic formula, and completing the square.

Factoring the Given Expression

Let's consider the expression (6x-1)(x-frac{1}{2})(2x-1)(x 7). To factor and simplify this expression, we need to break it down into its simplest form.

Step 1: Simplify Each Term

First, we will look at each term individually:

Notice that 2x-1 and x-frac{1}{2} can be written as:

Substituting these into the original expression, we get:

(6x-1)(x-frac{1}{2})2(x-frac{1}{2})(x 7)

Now, we can combine the like terms:

(6x-1)2(x-frac{1}{2})^2(x 7)

Step 2: Further Simplification

To further simplify, we can multiply the constants and the terms together:

(6x-1)2(x-frac{1}{2})^2(x 7) 12x(x-frac{1}{2})^2(x 7) - 2(x-frac{1}{2})^2(x 7)

Next, we can expand and simplify the expression:

12x(x^2 - x frac{1}{4})(x 7) - 2(x^2 - x frac{1}{4})(x 7)

This simplifies to:

12x(x^3 7x^2 - x^2 - 7x frac{1}{4}x 7frac{1}{4}) - 2(x^3 7x^2 - x^2 - 7x frac{1}{4}x 7frac{1}{4})

Further simplification gives:

12x^4 76x^3 - 14x^2 - 56x 17.5 - 2x^3 - 14x^2 2x^2 14x - 0.5x - 1.75

Combining like terms, we get:

12x^4 74x^3 - 26x^2 - 42x 16

Solving the Equation

Now, let's assume the equation equals a constant c:

(6x-1)(x-frac{1}{2})(2x-1)(x 7) c

We can now solve for x using the quadratic formula or by completing the square. Here, we will use the quadratic formula:

x frac{pm sqrt{b^2 - 4ac} - b}{2a}

Let's rewrite the equation in standard form:

x^2 frac{9}{8}x - frac{15}{16} - frac{c}{8} 0

Using the quadratic formula:

x frac{pm sqrt{left(frac{9}{16}right)^2 - 4 cdot 1 cdot left(-frac{15}{16} - frac{c}{8}right)} - frac{9}{16}}{2 cdot 1}

Further simplification gives:

x frac{pm sqrt{left(frac{81}{256}right) frac{60}{16} frac{c}{2}} - frac{9}{16}}{2}

Simplifying inside the square root:

x frac{pm sqrt{frac{324 900 16c}{256}} - frac{9}{16}}{2}

This simplifies to:

x frac{pm sqrt{frac{1224 16c}{256}} - frac{9}{16}}{2}

Finally, we get:

x frac{pm sqrt{frac{32132c}{256}} - frac{9}{16}}{2}

Therefore, for any value of c geq -frac{321}{32} -10.03125, the solutions are valid.

Conclusion

In conclusion, factoring and solving quadratic equations involves breaking down complex expressions into simpler forms, simplifying terms, and using the quadratic formula to find the solutions. Understanding these steps is crucial for solving a wide range of mathematical problems.

Keywords: factorization, quadratic equations, solving equations