Solving Quadratic Equations: A Comprehensive Guide with Step-by-Step Solutions

Quadratic equations are encountered in various real-world scenarios, from physics to engineering. A quadratic equation is an algebraic equation of the second degree, typically represented as ax2 bx c 0. In this article, we will walk through a step-by-step process to solve a specific quadratic equation: 6x2 - 5x - 4 0, using the quadratic formula.

Understanding the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is expressed as:

x [-b ± sqrt(b2 - 4ac)] / (2a)

where a, b, and c are coefficients from the original equation in the form ax2 bx c 0.

Solving the Quadratic Equation: 6x2 - 5x - 4 0

Let's break down the process of solving this equation using the quadratic formula.

Step 1: Identify the Coefficients

From the equation 6x2 - 5x - 4 0, we have:

a 6 b -5 c -4

Step 2: Calculate the Discriminant

The discriminant D is given by the formula:

D b2 - 4ac

Substitute the identified coefficients:

D (-5)2 - 4 * 6 * (-4)

D 25 96

D 121

Step 3: Apply the Quadratic Formula

Substitute a 6, b -5, and D 121 into the quadratic formula:

x [-(-5) ± sqrt(121)] / (2 * 6)

x (5 ± 11) / 12

Step 4: Simplify the Solutions

Calculate the two possible solutions:

For the positive root: x1 (5 11) / 12 16 / 12 4 / 3 For the negative root: x2 (5 - 11) / 12 -6 / 12 -1 / 2

Alternative Methods of Solving Quadratic Equations

There are several other methods to solve quadratic equations, including factoring and using a graphing calculator. Here are a couple of examples:

Factoring

In some cases, the quadratic equation can be factored directly. For example:

x2 - 4x - 5 0

This can be factored as:

(x - 5)(x 1) 0

Setting each factor to zero, we get:

x 5 or x -1

Using a Graphing Calculator

A graphing calculator can also be used to find the roots of the quadratic equation by determining the x-intercepts of the graph. Plotting the equation:

y 6x2 - 5x - 4

Observing the points where the parabola intersects the x-axis will give the roots.

Shortcut Methods and Tips

Becoming proficient with the quadratic formula can significantly reduce the time it takes to solve equations. Here are a couple of shortcut methods:

PQ-Formula

For an equation in the form ax2 bx c 0, the PQ-formula is given by:

x -b/2a ± sqrt((b/2a)2 - c/a)

For 6x2 - 5x - 4 0, apply the formula:

x -(-5)/2 * 6 ± sqrt((-5/2 * 6)2 - (-4/6))

After simplifying:

x 5/12 ± sqrt(25/144 4/6)

Identities and Patterns

Recognizing common patterns can help in quickly solving some quadratic equations. For example:

x2 - 4x - 5 (x - 5)(x 1)

This can be used to solve directly: