Solving Quadratic Equations: Techniques and Methods

Quadratic equations are fundamental in algebra, appearing in various fields such as physics, engineering, and economics. A quadratic equation is typically expressed in the form ax^2 bx c 0, where a, b, and c are constants, and a eq 0. This article will walk you through solving a specific quadratic equation and discuss the different methods to solve it, including the quadratic formula and factoring.

An Example Problem

Consider the quadratic equation x^2 - 4x 3 0.

Using the Quadratic Formula

The quadratic formula is one of the most direct methods to solve a quadratic equation. It is derived from the standard form of the quadratic equation ax^2 bx c 0 and is given by:

The solutions for x are:

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

For our equation x^2 - 4x 3 0, we have:

a 1 b -4 c 3

First, we calculate the discriminant D b^2 - 4ac:

D (-4)^2 - 4(1)(3) 16 - 12 4

Now, we apply the quadratic formula:

x frac{-(-4) pm sqrt{4}}{2(1)} frac{4 pm 2}{2}

This gives us two solutions:

x_1 frac{4 2}{2} 3 x_2 frac{4 - 2}{2} 1

Therefore, the solutions to the equation x^2 - 4x 3 0 are x 1 or x 3.

Factoring Method

Another common method to solve quadratic equations is by factoring. For the equation x^2 - 4x 3 0, we can rewrite the middle term to make it easier to factor:

x^2 - 4x 3 x^2 - 3x - x 3

Next, we group the terms:

x(x - 3) - 1(x - 3) (x - 1)(x - 3)

Now, we set each factor equal to zero:

x - 1 0 Rightarrow x 1 x - 3 0 Rightarrow x 3

Again, we find that the solutions are x 1 or x 3.

Discriminant Method

Another approach to solving quadratic equations is the discriminant method. The discriminant D can be used to determine the nature of the roots:

If D 0, the equation has two complex (imaginary) roots. If D 0, the equation has one real root (a repeated root). If D 0, the equation has two distinct real roots.

For our equation x^2 - 4x 3 0, the discriminant D 4, which is greater than 0, indicating that there are two distinct real roots.

The solutions can be obtained by:

Using the quadratic formula: x frac{-b pm sqrt{D}}{2a} Factoring: x^2 - 4x 3 (x - 1)(x - 3)

The final solutions are x 1 and x 3.

Conclusion

Quadratic equations can be solved using several methods, including the quadratic formula, factoring, and the discriminant. The choice of method can depend on the specific equation and the context of the problem. The factoring method is often preferred for simpler equations, while the quadratic formula offers a more general approach. The discriminant method provides insight into the nature of the roots.

By understanding and practicing these methods, you can solve quadratic equations with confidence and accuracy.