Solve $3z^2 - Z + 5 = 6$: Quadratic Equation Guide

Aug 6, 2025 by Rajiv Sharma 51 views

Solving Quadratic Equations: A Step-by-Step Guide to Finding Real Solutions

Hey guys! Today, we're diving into the exciting world of quadratic equations. These equations are like puzzles, and our mission is to find the values of the variable that make the equation true. We'll tackle the equation $3z^2 - z + 5 = 6$ and break down the process step-by-step so you can conquer any quadratic equation that comes your way. So, grab your thinking caps, and let's get started!

Understanding Quadratic Equations

Before we jump into solving, let's make sure we're all on the same page about what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. That might sound a little technical, but it just means the highest power of the variable (in our case, 'z') is 2. The general form of a quadratic equation is:

$ax^2 + bx + c = 0$

Where 'a', 'b', and 'c' are constants (numbers), and 'x' is the variable. Now, let's relate this to our equation: $3z^2 - z + 5 = 6$ . Notice that it looks similar, but it's not quite in the standard form because it doesn't equal zero. That's our first step – to rearrange the equation into the standard form.

Key characteristics of quadratic equations:

They have a term with the variable raised to the power of 2 ( $x^2$ or $z^2$ in our case).
They might have a term with the variable raised to the power of 1 (like 'x' or 'z').
They might have a constant term (a number without any variables).
The goal is to find the values of the variable that make the equation true, also known as the solutions or roots of the equation. These solutions represent the points where the parabola, which is the graph of the quadratic equation, intersects the x-axis. The number of real solutions can be zero, one, or two, depending on whether the parabola intersects the x-axis at zero, one, or two points, respectively.

Step 1: Rearrange the Equation into Standard Form

Our equation is $3z^2 - z + 5 = 6$ . To get it into the standard form ( $ax^2 + bx + c = 0$ ), we need to get rid of the '6' on the right side. How do we do that? Simple! We subtract 6 from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced.

$3z^2 - z + 5 - 6 = 6 - 6$

This simplifies to:

$3z^2 - z - 1 = 0$

Now, our equation is in the standard form. We can clearly see that:

a = 3
b = -1
c = -1

These values will be important when we use the quadratic formula later on. Understanding how to manipulate equations to achieve a standard form is a crucial skill in algebra. It not only prepares the equation for specific solution methods like the quadratic formula but also enhances our ability to recognize patterns and apply appropriate techniques. By consistently practicing this rearrangement, we build a solid foundation for tackling more complex algebraic problems.

Step 2: Choose a Method to Solve

Now that our equation is in standard form, we have a few options for solving it. The most common methods are:

Factoring: This involves breaking down the quadratic expression into two binomials. It's a great method when it works, but it's not always easy to factor quadratic equations.
Quadratic Formula: This is a foolproof method that always works, no matter how complicated the equation looks. It might seem a bit intimidating at first, but it's just a matter of plugging in the values of 'a', 'b', and 'c'.
Completing the Square: This method involves manipulating the equation to create a perfect square trinomial. It's a bit more involved than the other two methods, but it's useful for understanding the structure of quadratic equations.

For this particular equation, factoring might be a bit tricky. So, let's go with the quadratic formula. It's our trusty friend that always gets the job done.

Why the Quadratic Formula is a Universal Solver: The quadratic formula is derived from the process of completing the square on the general form of a quadratic equation, $ax^2 + bx + c = 0$ . This derivation ensures that the formula provides a solution for any quadratic equation, regardless of whether it can be easily factored or not. The formula encapsulates the relationship between the coefficients of the quadratic equation and its roots, making it a powerful tool in algebra.

Step 3: Apply the Quadratic Formula

The quadratic formula is given by:

$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Don't let it scare you! It looks complicated, but it's really just a matter of plugging in the values we identified earlier (a = 3, b = -1, c = -1). Let's do it step by step:

Substitute the values:

$z = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-1)}}{2(3)}$
Simplify:

$z = \frac{1 \pm \sqrt{1 + 12}}{6}$

$z = \frac{1 \pm \sqrt{13}}{6}$

Now we have two possible solutions, one with the plus sign and one with the minus sign:

$z_1 = \frac{1 + \sqrt{13}}{6}$
$z_2 = \frac{1 - \sqrt{13}}{6}$

The $\pm$ symbol in the quadratic formula is a crucial element that arises from the square root operation. When we solve for a variable that is squared, we inherently encounter two possible solutions: a positive and a negative root. This is because both the positive and negative values, when squared, yield the same positive result. The quadratic formula elegantly captures this duality, providing us with both potential solutions in a single expression.

Step 4: Simplify the Solutions (If Possible)

Our solutions are $z_1 = \frac{1 + \sqrt{13}}{6}$ and $z_2 = \frac{1 - \sqrt{13}}{6}$ . We need to simplify these and check if the radical $\sqrt{13}$ can be simplified further. Since 13 is a prime number, its square root cannot be simplified. Also, the fractions are already in their simplest form because there are no common factors between the numerator and denominator.

So, our solutions are already in the simplest form. We've done it!

Understanding the concept of simplifying solutions is fundamental in mathematics. It ensures that we express our answers in the most concise and understandable form. Simplifying radicals, like $\sqrt{13}$ in our example, involves checking for perfect square factors within the radicand (the number inside the square root). If there are no such factors, as in the case of 13, the radical is already in its simplest form. Similarly, simplifying fractions involves reducing them to their lowest terms by dividing both the numerator and the denominator by their greatest common divisor. These simplification skills are essential not only for solving quadratic equations but also for a wide range of mathematical problems.

Conclusion

We successfully solved the quadratic equation $3z^2 - z + 5 = 6$ and found the real solutions in simplest form:

$z_1 = \frac{1 + \sqrt{13}}{6}$
$z_2 = \frac{1 - \sqrt{13}}{6}$

We walked through the steps of rearranging the equation into standard form, choosing the quadratic formula as our method, applying the formula, and simplifying the solutions. Remember, practice makes perfect! The more you solve quadratic equations, the more comfortable you'll become with the process. So, keep practicing, and you'll be a quadratic equation-solving pro in no time!

By mastering the process of solving quadratic equations, we not only gain a valuable skill in algebra but also develop critical thinking and problem-solving abilities that are applicable across various fields. The quadratic formula, in particular, serves as a powerful tool that transcends the boundaries of mathematics, finding applications in physics, engineering, economics, and computer science. As we continue our mathematical journey, the skills and concepts learned in solving quadratic equations will undoubtedly serve as a solid foundation for tackling more advanced topics.