Understanding and Factoring the Quadratic Expression 3x^2 - x - 14
Quadratic expressions are fundamental in algebra and have wide applications in various fields, including physics, engineering, and economics. This article will guide you through the process of factoring the quadratic expression 3x^2 - x - 14 using both the quadratic formula and factoring methods.
Factoring Method for 3x^2 - x - 14
Let's start with the factoring method. The standard form of a quadratic expression is given by ax^2 bx c. For the expression 3x^2 - x - 14, we have a 3, b -1, and c -14.
First, identify the product of a and c, which is 3 × -14 -42. Now, find two factors of -42 that add up to b -1. The factors that satisfy this condition are 6 and -7, because 6 (-7) -1.
Using these factors, we can rewrite the middle term of the quadratic expression:
3x^2 6x - 7x - 14
Next, factor by grouping:
3x(x 2) - 7(x 2)
Now, factor out the common binomial term:
(3x - 7)(x - 2)
Therefore, the factored form of the expression 3x^2 - x - 14 is (3x - 7)(x - 2).
Using the Quadratic Formula
Another method to solve the quadratic equation is by using the quadratic formula:
x {-b ± √(b^2 - 4ac)} / 2a
Substitute the values of a 3, b -1, and c -14 into the formula:
x {-(-1) ± √((-1)^2 - 4(3)(-14))} / 2(3)
x {1 ± √(1 168)} / 6
x {1 ± √169} / 6
x {1 ± 13} / 6
This gives us two solutions:
x (1 13) / 6 14 / 6 7/3
x (1 - 13) / 6 -12 / 6 -2
So, the roots of the quadratic equation are x 7/3 and x -2. These roots correspond to the factors of the quadratic expression:
3x^2 - x - 14 3(x - 2)(x - 7/3)
Completing the Square Method
To factorize the quadratic expression 3x^2 - x - 14 using the completing the square method, follow these steps:
Multiply the coefficient of x^2 (which is 3) by the constant term (-14) to get -42. Find two factors of -42 that add up to the linear coefficient -1. These factors are 6 and -7. Split the middle term using these factors and then factor by grouping:3x^2 6x - 7x - 14
3x(x 2) - 7(x 2)
3(x 2)(x - 7/3)
This confirms that the factored form is 3(x - 2)(x - 7/3).
Conclusion
Both the quadratic formula and factoring methods can be used to factorize the quadratic expression 3x^2 - x - 14. The factors are (3x - 7)(x - 2), and the roots are x 7/3 and x -2.