Solving Quadratic Equations: Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of quadratic equations and finding the solution set for the equation $4x^2 + 13x = -9$. Don't worry, it might seem intimidating at first, but we'll break it down step by step. By the end of this guide, you'll be a pro at solving these types of problems. So, buckle up, and let's get started!

Understanding Quadratic Equations

First things first, what exactly is a quadratic equation? Well, a quadratic equation is a polynomial equation of the second degree, meaning it includes a term with $x$ raised to the power of 2 (i.e., $x^2$). The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ is not equal to 0. In our case, the equation $4x^2 + 13x = -9$ fits the bill, so we know we're dealing with a quadratic.

Quadratic equations can have two solutions, one solution, or no real solutions. The solutions, also known as roots, are the values of $x$ that make the equation true. These solutions can be found using several methods, including factoring, completing the square, and the quadratic formula. We'll explore each of these methods and discuss how to use them in practice.

Methods for Solving Quadratic Equations

1. Factoring

Factoring involves breaking down the quadratic expression into the product of two binomials. For this method to work, the quadratic expression must be factorable. Let's consider our original equation $4x^2 + 13x = -9$. First, we need to rewrite the equation in the standard form of $ax^2 + bx + c = 0$. So, we add 9 to both sides of the equation:

$4x^2 + 13x + 9 = 0$

Now, we need to find two numbers that multiply to give $ac$ (which is $4 * 9 = 36$) and add up to $b$ (which is 13). After some thought, the numbers 4 and 9 don't work; instead, the factors are 4 and 9. Rewrite the equation $4x^2 + 4x + 9x + 9 = 0$. Then, factor by grouping:

$4x(x + 1) + 9(x + 1) = 0$

$(4x + 9)(x + 1) = 0$

Now, set each factor equal to zero and solve for $x$:

$4x + 9 = 0$ or $x + 1 = 0$

$4x = -9$ or $x = -1$

x = -rac{9}{4} or $x = -1$

So, using factoring, the solution set is \left{-rac{9}{4}, -1\right}.

2. Completing the Square

Completing the square is a method that involves manipulating the quadratic equation to create a perfect square trinomial. This can be a bit more involved, but it's a reliable method that always works. Let's revisit our equation $4x^2 + 13x + 9 = 0$. First, divide the entire equation by the coefficient of $x^2$ (which is 4):

$x^2 + \frac{13}{4}x + \frac{9}{4} = 0$

Next, move the constant term to the right side of the equation:

$x^2 + \frac{13}{4}x = -\frac{9}{4}$

Now, take half of the coefficient of the $x$ term (which is $\frac{13}{4}$), square it $\left(\frac{13}{8}\right)^2 = \frac{169}{64}$, and add it to both sides of the equation:

$x^2 + \frac{13}{4}x + \frac{169}{64} = -\frac{9}{4} + \frac{169}{64}$

Rewrite the left side as a perfect square and simplify the right side:

$\left(x + \frac{13}{8}\right)^2 = \frac{169 - 144}{64}$

$\left(x + \frac{13}{8}\right)^2 = \frac{25}{64}$

Now, take the square root of both sides:

$x + \frac{13}{8} = \pm\frac{5}{8}$

Finally, solve for $x$:

$x = -\frac{13}{8} \pm \frac{5}{8}$

So, we have two solutions:

$x = -\frac{13}{8} + \frac{5}{8} = -1$

$x = -\frac{13}{8} - \frac{5}{8} = -\frac{18}{8} = -\frac{9}{4}$

Again, the solution set is \left{-rac{9}{4}, -1\right}.

3. The Quadratic Formula

Finally, we have the quadratic formula, a universal solution that can be used for any quadratic equation. The quadratic formula is derived from completing the square and is a quick way to find the solutions. The formula is:

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

For our equation, $4x^2 + 13x + 9 = 0$, we have $a = 4$, $b = 13$, and $c = 9$. Plug these values into the formula:

$x = \frac{-13 \pm \sqrt{13^2 - 4(4)(9)}}{2(4)}$

$x = \frac{-13 \pm \sqrt{169 - 144}}{8}$

$x = \frac{-13 \pm \sqrt{25}}{8}$

$x = \frac{-13 \pm 5}{8}$

So, we have two solutions:

$x = \frac{-13 + 5}{8} = \frac{-8}{8} = -1$

$x = \frac{-13 - 5}{8} = \frac{-18}{8} = -\frac{9}{4}$

Again, the solution set is \left{-rac{9}{4}, -1\right}.

Analyzing the Options

Now that we've solved the equation using multiple methods, let's look at the answer choices:

A. \left{-\frac{9}{4}, -1\right} - This is the correct solution set. B. $\{-9, -4\}$ - This is incorrect. C. \left{\frac{-13 - \sqrt{313}}{2}, \frac{-13 + \sqrt{313}}{2}\right} - This is incorrect. While it looks similar to the quadratic formula, the values are wrong. D. \left{\frac{-13 - \sqrt{313}}{8}, \frac{-13 + \sqrt{313}}{8}\right} - This is incorrect.

Conclusion

So, guys, the solution set for the equation $4x^2 + 13x = -9$ is \left{-\frac{9}{4}, -1\right}. We've successfully solved this equation using factoring, completing the square, and the quadratic formula, solidifying our understanding of quadratic equations. Keep practicing, and you'll become a master in no time! If you ever need help with math problems, don't hesitate to reach out. Practice makes perfect!