Solving the Equation 6x^2 - 7x - 3 3 Using the Factoring Method
In algebra, the factoring method is a powerful technique used to solve quadratic equations. This method is particularly useful when the equation can be factored into simpler expressions. In this article, we will explore how to solve the equation 6x^2 - 7x - 3 3 using the factoring method. We will step through the process, providing detailed explanations and examples to ensure a clear understanding of the concept.
Step-by-Step Guide
Let's begin with the given equation:
6x^2 - 7x - 3 3
The first step in solving this equation is to move all terms to one side of the equation, setting it equal to zero:
6x^2 - 7x - 3 - 3 0
6x^2 - 7x - 6 0
Multiplying and Finding Factors
To factor the quadratic equation, we start by multiplying the coefficient of the square term (6) by the constant term (-6), giving us:
6 × -6 -36
We then need to find two numbers that multiply to give -36 and add up to the coefficient of the middle term (-7). Through trial and error, we identify these numbers as -9 and 2. These can be verified as:
-9 × 2 -18
-9 2 -7
Splitting the Middle Term and Factoring
Using the identified numbers, we can split the middle term (-7x) into two parts:
6x^2 - 9x 2x - 6 0
Next, we factor the equation by grouping:
3x(2x - 3) 2(2x - 3) 0
(3x 2)(2x - 3) 0
The equation will be zero when the expressions in parentheses are equal to zero:
3x 2 0
2x - 3 0
Solving each of these for x:
3x 2 0
3x -2
x -2/3
2x - 3 0
2x 3
x 3/2
Thus, the solutions to the equation are:
x -2/3 or x 3/2
Alternative Method: Quadratic Formula
While the factoring method is efficient when the equation can be factored, the quadratic formula can provide a more straightforward approach:
x_{1,2} frac{-b pm sqrt{b^2-4ac}}{2a}
Substituting a 6, b -7, and c -6:
x_{1,2} frac{-(-7) pm sqrt{(-7)^2-4(6)(-6)}}{2(6)}
x_{1,2} frac{7 pm sqrt{49 144}}{12}
x_{1,2} frac{7 pm sqrt{193}}{12}
The solutions from the quadratic formula are approximately:
x_1 frac{7 - 11}{12} -frac{4}{12} -frac{1}{3}
x_2 frac{7 11}{12} frac{18}{12} frac{3}{2}
Both methods yield the same solutions:
x -frac{1}{3} or x frac{3}{2}
Conclusion
The factoring method is a valuable technique for solving quadratic equations, and it often provides a more intuitive understanding of the process. By mastering this method, you can solve equations more efficiently and gain deeper insights into algebraic expressions.