Solving Quadratics: A Step-by-Step Guide

Welcome! If you're wrestling with quadratic equations, you're in the right place. In this guide, we'll break down how to solve quadratic equations using the quadratic formula, step by simple step. We'll use the example equation $x^2 - 8x + 12 = 0$ to illustrate each stage, ensuring you grasp the method thoroughly. Whether you're a student tackling homework or just brushing up on your algebra, this explanation aims to make the process clear and straightforward. Let's dive in!

Understanding the Quadratic Formula

Before we apply the quadratic formula, let's understand what it is and why it’s so useful. A quadratic equation is generally expressed in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable we want to find. The quadratic formula is a universal tool that provides the solutions (also called roots or zeros) for any quadratic equation, regardless of whether the equation can be easily factored. This is incredibly valuable because not all quadratic equations can be factored neatly.

The quadratic formula itself is:

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

This formula tells us that the solutions for $x$ can be found by plugging in the values of $a$, $b$, and $c$ from our quadratic equation into the formula. The $\pm$ symbol indicates that there are typically two solutions: one where we add the square root term and one where we subtract it. These solutions represent the points where the parabola described by the quadratic equation intersects the x-axis.

Why is this formula so powerful? Because it works every time! Whether your coefficients are integers, fractions, or even decimals, the quadratic formula will provide the correct solutions, assuming they exist. It bypasses the need to guess factors or complete the square, offering a direct route to the answer. This makes it an indispensable tool in algebra and beyond.

For example, consider the equation $2x^2 + 5x - 3 = 0$. Here, $a = 2$, $b = 5$, and $c = -3$. Plugging these values into the quadratic formula will give us the solutions for $x$. We'll go through the steps in detail later, but for now, understand that the quadratic formula is the key to unlocking the solutions of any quadratic equation.

Understanding the formula's components is also crucial. The term inside the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant tells us about the nature of the solutions: if it's positive, there are two distinct real solutions; if it's zero, there is exactly one real solution (a repeated root); and if it's negative, there are two complex solutions. This insight can save time and help anticipate the type of answers you'll get.

In summary, the quadratic formula is an essential tool for solving quadratic equations. It provides a reliable and direct method for finding the solutions, regardless of the complexity of the equation. By understanding the formula and its components, you'll be well-equipped to tackle any quadratic equation that comes your way.

Step 1: Identify a, b, and c

The first step in using the quadratic formula is to correctly identify the values of $a$, $b$, and $c$ from the quadratic equation. Remember, the standard form of a quadratic equation is $ax^2 + bx + c = 0$. The coefficients $a$, $b$, and $c$ are the numbers that multiply $x^2$, $x$, and the constant term, respectively.

In our example equation, $x^2 - 8x + 12 = 0$, we can identify the coefficients as follows:

• $a$ is the coefficient of $x^2$. Since $x^2$ is the same as $1x^2$, we have $a = 1$.
• $b$ is the coefficient of $x$. In this case, $b = -8$. Pay careful attention to the sign!
• $c$ is the constant term. Here, $c = 12$.

It’s crucial to get these values right because they will be plugged into the quadratic formula. A mistake in identifying $a$, $b$, or $c$ will lead to incorrect solutions. Let’s look at some more examples to practice identifying these coefficients.

Consider the equation $3x^2 + 5x - 2 = 0$. Here, $a = 3$, $b = 5$, and $c = -2$. Notice again the importance of the sign for $c$.

Another example is $x^2 - 4 = 0$. In this case, $a = 1$, $b = 0$ (since there is no $x$ term), and $c = -4$. This highlights that if a term is missing, its coefficient is zero.

What about the equation $2x^2 - 7x = 0$? Here, $a = 2$, $b = -7$, and $c = 0$ (since there is no constant term).

Once you become comfortable identifying $a$, $b$, and $c$, you’ll find that plugging them into the quadratic formula becomes much easier. This step is fundamental to correctly solving quadratic equations using this method.

In summary, take your time to carefully identify the coefficients $a$, $b$, and $c$ from the given quadratic equation. Remember to pay attention to the signs and include zero as the coefficient if a term is missing. This will set you up for success in the subsequent steps.

Step 2: Plug the Values into the Quadratic Formula

Now that we've identified $a$, $b$, and $c$ for our equation $x^2 - 8x + 12 = 0$, where $a = 1$, $b = -8$, and $c = 12$, we can plug these values into the quadratic formula:

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

Substituting the values, we get:

$x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(1)(12)}}{2(1)}$

Notice how each value has been carefully placed into the formula. The negative sign in front of $b$ is part of the formula and must be included. Also, make sure to use parentheses when substituting negative numbers to avoid sign errors.

Let’s simplify this expression step by step. First, $-(-8)$ becomes $8$, so we have:

$x = \frac{8 \pm \sqrt{(-8)^2 - 4(1)(12)}}{2(1)}$

Next, let's simplify the expression under the square root. $(-8)^2$ is $64$, and $4(1)(12)$ is $48$. So the expression becomes:

$x = \frac{8 \pm \sqrt{64 - 48}}{2(1)}$

Now, we subtract $48$ from $64$:

$x = \frac{8 \pm \sqrt{16}}{2(1)}$

Finally, $2(1)$ simplifies to $2$, so we have:

$x = \frac{8 \pm \sqrt{16}}{2}$

Plugging values into the quadratic formula requires careful attention to detail. Double-check each substitution to ensure accuracy. The order of operations (PEMDAS/BODMAS) is also important when simplifying the expression. By taking it one step at a time and being methodical, you can avoid common errors and set yourself up for finding the correct solutions.

In summary, substituting the values of $a$, $b$, and $c$ into the quadratic formula is a critical step. Ensure you substitute correctly, paying close attention to signs and parentheses. Then, simplify the expression following the order of operations. This careful approach will lead to accurate results.

Step 3: Simplify and Solve for x

Continuing from where we left off, we have:

$x = \frac{8 \pm \sqrt{16}}{2}$

Now we need to simplify the square root. The square root of $16$ is $4$, so the equation becomes:

$x = \frac{8 \pm 4}{2}$

This gives us two possible solutions for $x$, one where we add $4$ and one where we subtract $4$. Let’s solve for each case separately.

Case 1: Adding

$x = \frac{8 + 4}{2}$

$x = \frac{12}{2}$

$x = 6$

So, one solution is $x = 6$.

Case 2: Subtracting

$x = \frac{8 - 4}{2}$

$x = \frac{4}{2}$

$x = 2$

So, the other solution is $x = 2$.

Therefore, the solutions to the quadratic equation $x^2 - 8x + 12 = 0$ are $x = 6$ and $x = 2$.

To verify these solutions, you can plug them back into the original equation to see if they satisfy it. Let's check:

For $x = 6$:

$(6)^2 - 8(6) + 12 = 36 - 48 + 12 = 0$

For $x = 2$:

$(2)^2 - 8(2) + 12 = 4 - 16 + 12 = 0$

Both solutions satisfy the equation, so they are correct.

Simplifying and solving for $x$ involves breaking down the expression obtained from the quadratic formula into two separate cases: one with addition and one with subtraction. Each case is then simplified to find the value of $x$. This step requires careful arithmetic and attention to detail to ensure accurate solutions.

In summary, simplify the square root, then separate the equation into two cases using the $\pm$ sign. Solve each case independently to find the two possible values of $x$. Finally, verify your solutions by plugging them back into the original equation to ensure they are correct.

Conclusion

Congratulations! You've successfully used the quadratic formula to solve the equation $x^2 - 8x + 12 = 0$. By following these steps—identifying $a$, $b$, and $c$, plugging the values into the formula, and simplifying to solve for $x$—you can tackle any quadratic equation. Remember to take your time, double-check your work, and practice regularly to master this essential algebraic skill.

The quadratic formula is a powerful tool that provides a systematic way to find the solutions to any quadratic equation, regardless of its complexity. Understanding how to use it effectively is a valuable skill in mathematics and can be applied in various fields, including physics, engineering, and computer science. Keep practicing, and you'll become proficient in solving quadratic equations with ease!

For further learning and practice, check out resources like Khan Academy's quadratic equation section to deepen your understanding and skills.