Solving For Roots: 2x - 3 = -5x² Equation Explained

Hey there, math enthusiasts! Today, we're diving into a quadratic equation to find its roots. Roots, in simple terms, are the values of 'x' that make the equation true. We'll be tackling the equation $2x - 3 = -5x^2$, and by the end of this guide, you'll know exactly how to solve it and select the correct answers.

Understanding the Problem

The equation we're dealing with is $2x - 3 = -5x^2$. This looks a bit jumbled, right? The first step is to rearrange it into the standard quadratic form, which is $ax^2 + bx + c = 0$. This form makes it much easier to apply our solving methods. Think of it like organizing your toolbox before starting a project; having everything in order helps a lot.

Rearranging the Equation

To get our equation into the standard form, we need to move all the terms to one side, leaving zero on the other. We can do this by adding $5x^2$ to both sides of the equation:

$2x - 3 + 5x^2 = -5x^2 + 5x^2$

This simplifies to:

$5x^2 + 2x - 3 = 0$

Now we have a quadratic equation in the standard form, where $a = 5$, $b = 2$, and $c = -3$. See how much cleaner it looks? This is our starting point for finding the roots.

Methods to Find the Roots

There are a couple of ways we can go about finding the roots of a quadratic equation: factoring and using the quadratic formula. Let's explore both, as each has its strengths and can be useful in different situations.

1. Factoring the Quadratic Equation

Factoring is like reverse multiplication. We're trying to find two binomials (expressions with two terms) that, when multiplied together, give us our quadratic equation. If you're comfortable with factoring, this can be a quick method. However, not all quadratic equations are easily factorable, so it's good to have other tools in your arsenal.

In our case, we need to find two binomials that multiply to give $5x^2 + 2x - 3$. This might take a bit of trial and error. We're looking for factors of $5x^2$ and -3 that, when combined in the right way, give us the middle term, $2x$.

After some thought, we can factor the equation as follows:

$(5x - 3)(x + 1) = 0$

Why does this work? If you multiply out $(5x - 3)(x + 1)$, you'll see that it indeed equals $5x^2 + 2x - 3$. Factoring is a bit like solving a puzzle; it's satisfying when the pieces click into place.

2. Using the Quadratic Formula

When factoring doesn't come easily or the equation is not factorable, the quadratic formula is your best friend. This formula works for any quadratic equation in the standard form and gives you the roots directly. It might look intimidating at first, but once you get the hang of plugging in the values, it's quite straightforward.

The quadratic formula is:

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

Remember those values we identified earlier ($a = 5$, $b = 2$, and $c = -3$)? Now we plug them into the formula:

$x = \frac{-2 ± \sqrt{2^2 - 4(5)(-3)}}{2(5)}$

Let's simplify this step by step:

$x = \frac{-2 ± \sqrt{4 + 60}}{10}$

$x = \frac{-2 ± \sqrt{64}}{10}$

$x = \frac{-2 ± 8}{10}$

Now we have two possible solutions for x, one with the plus sign and one with the minus sign. Let's calculate them.

Calculating the Roots

Now that we've factored the equation and set up the quadratic formula, it's time to actually calculate the roots. This is where the numerical answers come into play. We'll use both the factored form and the quadratic formula results to verify our solutions.

Roots from Factoring

From the factored form $(5x - 3)(x + 1) = 0$, we can find the roots by setting each factor equal to zero:

$5x - 3 = 0$ or $x + 1 = 0$

Solving the first equation:

$5x = 3$

$x = \frac{3}{5}$

Solving the second equation:

$x = -1$

So, from factoring, we found the roots to be $x = \frac{3}{5}$ and $x = -1$. These are our potential answers.

Roots from the Quadratic Formula

From the quadratic formula, we had two possibilities:

$x = \frac{-2 + 8}{10}$ and $x = \frac{-2 - 8}{10}$

Let's calculate each one:

$x = \frac{6}{10} = \frac{3}{5}$

$x = \frac{-10}{10} = -1$

Notice anything? The roots we found using the quadratic formula are the same as those we found by factoring! This is a great way to double-check your work. If you get different results from different methods, it's a sign to go back and look for a mistake.

Selecting the Correct Values

Now that we've calculated the roots using both factoring and the quadratic formula, we have a solid confirmation of our solutions. The roots are $x = \frac{3}{5}$ and $x = -1$. Let's look back at the original options provided to select the correct ones.

The options given were:

A. $x = -\frac{2}{3}$

B. $x = -1$

C. $x = \frac{3}{5}$

D. $x = \frac{5}{3}$

Comparing our calculated roots with the options, we can see that:

Option B, $x = -1$, matches one of our roots.

Option C, $x = \frac{3}{5}$, matches our other root.

Therefore, the two values of x that are roots of the equation $2x - 3 = -5x^2$ are $x = -1$ and $x = \frac{3}{5}$. We've successfully identified the correct answers!

Conclusion

We've walked through the entire process of solving a quadratic equation, from rearranging it into standard form to using factoring and the quadratic formula to find the roots. Remember, the key is to understand the steps and apply them methodically. Whether you prefer factoring or the quadratic formula, having both methods in your toolkit is a great advantage.

Finding the roots of equations might seem daunting at first, but with practice, it becomes second nature. Keep practicing, and you'll be solving quadratic equations like a pro in no time! And remember, Khan Academy is an excellent resource for further learning and practice in algebra.