Solving the Equation 2x^2 - 5x - 63 0: A Step-by-Step Guide

In this article, we will walk through the process of solving the equation 2x^2 - 5x - 63 0. This type of equation is known as a quadratic equation, and it is a fundamental concept in algebra. Understanding how to solve such equations is crucial for various applications in mathematics, science, and engineering. Let's dive in and explore the solution in detail.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. It can be written in the standard form (ax^2 bx c 0), where (a), (b), and (c) are constants, and (a eq 0). The solutions to a quadratic equation can be found using the quadratic formula:

x frac{?b ± sqrt{b^2 - 4ac}}{2a}

Given Problem and Equation

The given equation is:

2x^2 - 5x - 63 0

Here, the coefficients are a 2, b -5, and c -63.

Solving the Quadratic Equation

We will solve the equation step by step using the quadratic formula.

Step 1: Identify the Coefficients

First, identify the values of a, b, and c from the given equation:

a 2

b -5

c -63

Step 2: Calculate the Discriminant

The discriminant of a quadratic equation is given by the formula:

delta b^2 - 4ac

Substitute the values of a, b, and c into the discriminant formula:

(-5)^2 - 4(2)(-63) 25 504 529

Since the discriminant is positive, there are two distinct real roots.

Step 3: Apply the Quadratic Formula

Now, we can use the quadratic formula to find the solutions:

x frac{?b ± sqrt{delta}}{2a} frac{5 ± sqrt{529}}{4} frac{5 ± 23}{4}

This gives us two possible solutions:

x frac{5 23}{4} frac{28}{4} 7

x frac{5 - 23}{4} frac{-18}{4} -4.5

Verification

To ensure the accuracy of the solutions, let's substitute them back into the original equation.

Verification for x 7

Substitute x 7 into the equation:

2(7)^2 - 5(7) - 63 2(49) - 35 - 63 98 - 35 - 63 63

The equation holds true, so x 7 is a valid solution.

Verification for x -4.5

Substitute x -4.5 into the equation:

2(-4.5)^2 - 5(-4.5) - 63 2(20.25) 22.5 - 63 40.5 22.5 - 63 63

The equation also holds true, so x -4.5 is a valid solution.

Conclusion

The solutions to the quadratic equation 2x^2 - 5x - 63 0 are:

boxed{7} and boxed{-4.5}

The integer solution to the problem is boxed{7}, which is the most practical and commonly expected result in many real-world applications.

Additional Insights

Solving quadratic equations is not only useful in mathematical contexts but also in various real-life scenarios, such as calculating dimensions, analyzing financial investments, and understanding physical phenomena. Understanding the quadratic formula and its application is a fundamental skill that every student and professional in fields involving mathematical modeling should master.

For further practice or if you encounter similar problems, consider using graphing software or calculators to visualize the solutions and their practical implications. The process of solving quadratic equations is a key building block in advanced mathematics, and proficiency in this area will greatly enhance your problem-solving abilities in multiple disciplines.