Solving a Quadratic Equation: The 3x2 - 4x - 1 0 Example

Quadratic equations are an essential tool in algebra. They help us solve problems involving quadratic functions, which are commonly found in various real-world scenarios. In this article, we will explore how to solve the quadratic equation 3x2 - 4x - 1 0 using multiple methods, including the quadratic formula and factoring. Understanding these methods will enhance your algebraic skills and prepare you for more complex mathematical problems.

Solving by the Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation in the form ax2 bx c 0. For the equation 3x2 - 4x - 1 0, the coefficients are:

The quadratic formula is given by

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

Let's substitute the values:

[x frac{-(-4) pm sqrt{(-4)^2 - 4(3)(-1)}}{2(3)}]

Simplifying the expression inside the square root:

[x frac{4 pm sqrt{16 12}}{6}]

[x frac{4 pm sqrt{28}}{6}]

[x frac{4 pm 2sqrt{7}}{6}]

Simplifying further:

[x frac{2 pm sqrt{7}}{3}]

We find the two solutions:

[x frac{2 sqrt{7}}{3}]

[x frac{2 - sqrt{7}}{3}]

Solving by Factoring

Factoring is a method that involves breaking down the quadratic equation into simpler expressions. For our equation 3x2 - 4x - 1 0, we need to find two numbers that multiply to 3(-1) -3 and add up to -4.

The pairs that satisfy this condition are -3 and 1. Using these, we can rewrite the middle term and factor the quadratic expression:

[3x^2 - 3x - x - 1 0]

We then group the terms:

[3x(x - 1) - 1(x - 1) 0]

Factoring out the common term:

[(3x - 1)(x - 1) 0]

Setting each factor to zero:

[3x - 1 0 rightarrow x frac{1}{3}]

[x - 1 0 rightarrow x 1]

The solutions are the same as those obtained using the quadratic formula.

Alternative Methods

Besides the quadratic formula and factoring, there are other methods to solve quadratic equations. One such method involves completing the square:

[3x^2 - 4x - 1 0]

Factor out the 3:

[3(x^2 - frac{4}{3}x) - 1 0]

Add and subtract the square of half the coefficient of x inside the parenthesis:

[3(x^2 - frac{4}{3}x frac{4}{9} - frac{4}{9}) - 1 0]

[3((x - frac{2}{3})^2 - frac{4}{9}) - 1 0]

[3(x - frac{2}{3})^2 - frac{4}{3} - 1 0]

[3(x - frac{2}{3})^2 - frac{7}{3} 0]

[3(x - frac{2}{3})^2 frac{7}{3}]

[(x - frac{2}{3})^2 frac{7}{9}]

[x - frac{2}{3} pm sqrt{frac{7}{9}}]

[x frac{2}{3} pm frac{sqrt{7}}{3}]

[x frac{2 pm sqrt{7}}{3}]

This confirms the solutions obtained by the quadratic formula and factoring methods.

Conclusion

Understanding and applying different methods to solve quadratic equations is crucial. The quadratic formula and factoring are two effective methods that work for a wide range of equations. Additionally, methods like completing the square offer alternative ways to approach and solve these equations. Mastering these techniques will not only improve your algebraic skills but also enhance your problem-solving capabilities in various mathematical contexts.