Solving Quadratic Equations Using the Quadratic Formula: A Comprehensive Guide

In this comprehensive guide, we will walk through the process of solving quadratic equations using the quadratic formula. The quadratic formula is a powerful tool for finding the roots of a quadratic equation. This method is particularly useful when the equation is not easily factorable or when you need to find the exact roots.

Understanding the Quadratic Formula

The quadratic formula is given by:

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

Detailed Steps to Solve a Quadratic Equation

Let's solve the given equation step-by-step:

Step 1: Expand and Rearrange the Equation

The given equation is:

2x^2 xx - 14 - 5

First, expand and rearrange the equation:

2x^2 4x^2

xx - 14 - 5 x^2 - 14x - 5

So, the equation becomes:

4x^2 x^2 - 14x - 5

Move all terms to one side to set the equation to zero:

4x^2 - x^2 - 14x - 5 0

Combine like terms:

3x^2 - 14x - 5 0

Step 2: Identify Coefficients

In the standard form ax^2 bx c 0, the coefficients are:

a 3

b -14

c -5

Step 3: Use the Quadratic Formula

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

x (frac{{-(-14) pm sqrt{{(-14)^2 - 4 cdot 3 cdot (-5)}}}}{2 cdot 3})

Calculate the discriminant:

(-14)^2 196

4 cdot 3 cdot (-5) -60

b^2 - 4ac 196 - (-60) 256

Substitute the discriminant back into the formula:

x (frac{{14 pm sqrt{{256}}}}{6})

Simplify (sqrt{256}):

(sqrt{{256}} 16)

Substitute 16 back into the formula:

x (frac{{14 pm 16}}{6})

Step 4: Further Simplify

Divide all terms in the numerator by 2:

x (frac{{7 pm 8}}{3})

Thus, the solutions are:

x (frac{{7 8}}{3} frac{{15}}{3} 5)

x (frac{{7 - 8}}{3} (frac{{-1}}{3} approx -0.333)

General Approach to Solving Quadratic Equations

When solving a quadratic equation, you first need to normalize your equation by bringing it into the form:

(displaystyle x^2 px q 0)

(displaystyle ax^2 bx c 0)

Depending on which form you use, the solutions can be found as:

(displaystyle x_{12} -frac{p}{2} pm sqrt{frac{p^2}{4} - q})

(displaystyle x_{12} frac{{-b pm sqrt{{b^2 - 4ac}}}}{2a})

In both cases, the discriminant ((Delta b^2 - 4ac)) provides valuable information about the nature of the solutions:

If (Delta > 0), you will get two real solutions. If (Delta 0), you will get one real solution. If (Delta

Conclusion

By following these steps, you can solve any quadratic equation using the quadratic formula. Mastering this method not only helps in solving equations but also enhances your understanding of algebraic concepts.