Solving $-8y^2 - 2 = 3y$ With The Quadratic Formula

Let's dive into the world of quadratic equations! In this comprehensive guide, we'll tackle the equation $-8y^2 - 2 = 3y$ using the powerful quadratic formula. Whether you're a student brushing up on algebra or just curious about math, this breakdown will help you understand each step clearly.

What is a Quadratic Equation?

Before we jump into solving, let's define what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (usually x or y) is 2. The standard form of a quadratic equation is:

$ax^2 + bx + c = 0$

Where a, b, and c are constants, and a is not equal to zero. If a were zero, the equation would become linear, not quadratic.

Key Features of Quadratic Equations:

• The squared term: The presence of a term like $ax^2$ is what makes it quadratic.
• The coefficients: The values a, b, and c are called coefficients. a is the coefficient of the squared term, b is the coefficient of the linear term, and c is the constant term.
• Solutions or roots: Quadratic equations can have up to two distinct solutions, also known as roots or zeros. These are the values of the variable that make the equation true.

Why Learn About Quadratic Equations?

Quadratic equations pop up in many areas of mathematics, science, and engineering. They can model a variety of real-world situations, such as the trajectory of a projectile, the shape of a suspension bridge, or the growth of a population. Understanding how to solve them is a fundamental skill in these fields.

Understanding the Quadratic Formula

Now, let's introduce the star of our show: the quadratic formula. This formula provides a direct method for finding the solutions (roots) of any quadratic equation. It's derived from the method of completing the square and is a reliable tool for solving equations that are difficult or impossible to factor.

The quadratic formula is given by:

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

Where a, b, and c are the coefficients from the standard form of the quadratic equation ($ax^2 + bx + c = 0$).

Let's break down the formula:

• The $\pm$ symbol: This symbol means "plus or minus." It indicates that there are potentially two solutions, one found by adding the square root term and the other by subtracting it.
• The square root: The expression under the square root, $b^2 - 4ac$, is called the discriminant. The discriminant tells us about the nature of the solutions:
  • If $b^2 - 4ac > 0$, there are two distinct real solutions.
  • If $b^2 - 4ac = 0$, there is exactly one real solution (a repeated root).
  • If $b^2 - 4ac < 0$, there are two complex solutions.
• The denominator: The entire numerator is divided by $2a$.

When to Use the Quadratic Formula

The quadratic formula is a versatile tool, but it's especially useful in these situations:

• When factoring is difficult: Some quadratic equations are hard or impossible to factor using traditional methods.
• When you need a guaranteed solution: The quadratic formula always provides a solution, whether it's a real number or a complex number.
• When the equation is already in standard form: If your equation is in the form $ax^2 + bx + c = 0$, the quadratic formula is ready to be applied.

Solving $-8y^2 - 2 = 3y$ Using the Quadratic Formula: A Step-by-Step Solution

Now, let’s get our hands dirty and solve the equation $-8y^2 - 2 = 3y$ using the quadratic formula. We'll break it down into manageable steps to ensure clarity.

Step 1: Rewrite the Equation in Standard Form

As we discussed earlier, the standard form of a quadratic equation is $ax^2 + bx + c = 0$. Our first task is to rearrange the given equation into this form. The given equation is:

$-8y^2 - 2 = 3y$

To get it into standard form, we need to move the $3y$ term to the left side of the equation. We can do this by subtracting $3y$ from both sides:

$-8y^2 - 3y - 2 = 0$

Now, we have our equation in the standard form, with $a = -8$, $b = -3$, and $c = -2$.

Step 2: Identify the Coefficients a, b, and c

This step is crucial because we'll need these values to plug into the quadratic formula. From our standard form equation, $-8y^2 - 3y - 2 = 0$, we can identify the coefficients as follows:

• $a = -8$
• $b = -3$
• $c = -2$

Double-checking these values is always a good idea to avoid errors in the next steps.

Step 3: Plug the Coefficients into the Quadratic Formula

Now comes the exciting part! We'll take the values we identified and substitute them into the quadratic formula:

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

Plugging in our values, we get:

$y = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(-8)(-2)}}{2(-8)}$

Notice how we've carefully substituted each coefficient into its place in the formula. It's important to pay attention to signs, especially when dealing with negative numbers.

Step 4: Simplify the Expression

Now we need to simplify the expression we obtained in the previous step. This involves performing the arithmetic operations both inside the square root and in the rest of the formula.

First, let's simplify inside the square root:

$(-3)^2 - 4(-8)(-2) = 9 - 64 = -55$

So our equation now looks like this:

$y = \frac{3 \pm \sqrt{-55}}{-16}$

Notice that we have a negative number inside the square root. This tells us that the solutions will be complex numbers.

Step 5: Express the Solution in Complex Form

Since we have a negative number under the square root, we need to express the solution using the imaginary unit, i, where $i = \sqrt{-1}$. We can rewrite $\sqrt{-55}$ as:

$\sqrt{-55} = \sqrt{55} \cdot \sqrt{-1} = \sqrt{55}i$

Now we can rewrite our equation as:

$y = \frac{3 \pm \sqrt{55}i}{-16}$

To express the solution in standard complex form ($a + bi$), we can divide both the real and imaginary parts of the numerator by the denominator:

$y = \frac{3}{-16} \pm \frac{\sqrt{55}i}{-16}$

$y = -\frac{3}{16} \pm \frac{\sqrt{55}}{16}i$

Thus, we have two complex solutions:

$y_1 = -\frac{3}{16} + \frac{\sqrt{55}}{16}i$

$y_2 = -\frac{3}{16} - \frac{\sqrt{55}}{16}i$

Step 6: Verify the Solution (Optional but Recommended)

To ensure our solution is correct, we can plug the values we found back into the original equation. This can be a bit tedious with complex numbers, but it's a good way to check for errors. For brevity, we'll skip the verification step here, but it's an important step in practice.

Conclusion

Congratulations! You've successfully solved the quadratic equation $-8y^2 - 2 = 3y$ using the quadratic formula. We walked through each step, from rewriting the equation in standard form to expressing the solution in complex form. Remember, the quadratic formula is a powerful tool for solving any quadratic equation, and with practice, you'll become more comfortable using it.

Key Takeaways

• The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• The discriminant ($b^2 - 4ac$) tells us about the nature of the solutions.
• Complex solutions arise when the discriminant is negative.

Solving quadratic equations is a fundamental skill in mathematics. By understanding the quadratic formula and practicing its application, you'll be well-equipped to tackle a wide range of problems in algebra and beyond.

For further exploration of quadratic equations and the quadratic formula, consider visiting resources like Khan Academy's Quadratic Equations section.