Understanding Quadratic Equations and Their Roots

Quadratic equations are a fundamental part of algebra and appear in many real-world applications. A quadratic equation is an equation of the second degree in one variable, typically written in the form:

ax^2 bx c 0

This article will explore the roots of the quadratic equation 2x^2 3x - 4 0 and provide a comprehensive guide on how to solve such equations using the quadratic formula and factoring.

Solving the Quadratic Equation 2x^2 3x - 4 0

Given the quadratic equation 2x^2 3x - 4 0, we can identify the coefficients as:

a 2 b 3 c -4

One method to solve a quadratic equation is by using the quadratic formula, which is given by:

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

First, let's calculate the determinant b^2 - 4ac:

b^2 - 4ac 3^2 - 4(2)(-4) 9 32 41

Now, we can substitute the values of a, b, and the determinant into the quadratic formula:

x frac{-3 pm sqrt{41}}{2(2)} frac{-3 pm sqrt{41}}{4}

This yields two roots:

x_1 frac{-3 sqrt{41}}{4} approx 0.85 x_2 frac{-3 - sqrt{41}}{4} approx -2.35

Thus, the roots of the equation 2x^2 3x - 4 0 are approximately 0.85 and -2.35.

Graphical Interpretation

A quadratic equation can be graphed as a parabola. The graph of 2x^2 3x - 4 0 will intersect the x-axis at the roots x 0.85 and x -2.35. Here is a graphical representation of the equation:

Graph of the equation 2x^2 3x - 4 0

Alternative Methods for Solving Quadratic Equations

An alternative method to solving quadratic equations is factoring. Let's consider the equation:

x^2 3x - 4 0

Factoring this equation requires finding two numbers that multiply to -4 and add to 3. These numbers are -1 and 4. Therefore, we can rewrite the equation as:

x^2 4x - x - 4 0

Cancelling out the common term x 4, we get:

(x 4)(x - 1) 0

Setting each factor equal to zero, we find the roots:

x 4 0 implies x -4 x - 1 0 implies x 1

Thus, the roots of the equation x^2 3x - 4 0 are -4 and 1.

Summary

Quadratic equations can be solved using various methods, including the quadratic formula and factoring. The roots of a quadratic equation are the values of x that satisfy the equation. Understanding these methods is crucial for solving a wide range of algebraic problems and real-world applications.

Conclusion

Quadratic equations and their roots play a vital role in mathematics and its applications. By mastering the techniques for solving quadratic equations, you can unlock a deeper understanding of algebra and its practical implications.