Unlocking Quadratic Equations: A Step-by-Step Guide

Welcome! Let's dive into the fascinating world of quadratic equations. We'll explore how to solve them through factoring, a fundamental skill in algebra. Factoring is like detective work, breaking down complex expressions into simpler components. This approach lets us find the roots or solutions, which are the values of x that make the equation true. We'll be working through several examples, breaking down each step to make the process clear and easy to follow. Get ready to flex your mathematical muscles and unlock the secrets of quadratic equations! This guide is designed to transform complex problems into manageable steps, making the learning process both effective and enjoyable. Remember, practice is key, so grab a pen and paper, and let's get started. We will start with the first equation which is $16x^2 - 8x + 1 = 0$, then we will move to the second one, which is $x^2 - x - 30 = 0$. After that, we'll solve $9x^2 - 49 = 0$ and finally, we'll address the quadratic equation $3x^2 + 17x - 6 = 0$. By the end of this guide, you will be well-equipped to tackle similar problems with confidence.

Solving $16x^2 - 8x + 1 = 0$: Perfect Square Trinomial

Let's begin with the equation $16x^2 - 8x + 1 = 0$. This is a classic example of a perfect square trinomial. Recognizing this pattern is the key to a quick and efficient solution. In a perfect square trinomial, the first and last terms are perfect squares, and the middle term is twice the product of the square roots of the first and last terms. In our equation, the square root of $16x^2$ is $4x$, and the square root of $1$ is $1$. Twice the product of $4x$ and $1$ is $8x$, which matches the middle term (ignoring the negative sign for now, as it indicates a subtraction). So, this equation fits the pattern perfectly. Therefore, we can factor it as $(4x - 1)^2 = 0$. To solve for x, we take the square root of both sides, resulting in $4x - 1 = 0$. Then, we add $1$ to both sides, which gives us $4x = 1$. Finally, we divide both sides by $4$ to isolate x, giving us x = rac{1}{4}. Thus, the solution to the equation $16x^2 - 8x + 1 = 0$ is x = rac{1}{4}. This means that the quadratic equation has one real root, also known as a repeated root, because the factor $(4x - 1)$ appears twice. This root represents the point where the parabola (the graph of the quadratic equation) touches the x-axis.

Now, let's break down the steps clearly.

1. Recognize the Pattern: Identify that $16x^2 - 8x + 1$ is a perfect square trinomial.
2. Factor: Factor the expression into $(4x - 1)^2 = 0$.
3. Solve: Set the factor equal to zero: $4x - 1 = 0$.
4. Isolate x: Add $1$ to both sides: $4x = 1$.
5. Find x: Divide both sides by $4$: x = rac{1}{4}.

So, the solution to the equation is x = rac{1}{4}. Understanding perfect square trinomials allows you to solve this kind of equation quickly and efficiently. Keep in mind this pattern as it often appears in algebra problems.

Solving $x^2 - x - 30 = 0$: Factoring Quadratics

Next, let's solve $x^2 - x - 30 = 0$. This equation requires a different factoring approach. Here, we're looking for two numbers that multiply to give us the constant term (-30) and add up to the coefficient of the x term (-1). Think of it like this: $(x + a)(x + b) = x^2 + (a + b)x + ab$, where a and b are the two numbers we're looking for. In this case, those two numbers are $-6$ and $5$, because $-6 * 5 = -30$ and $-6 + 5 = -1$. Therefore, we can factor the equation as $(x - 6)(x + 5) = 0$. Now, we set each factor equal to zero and solve for x. For the first factor, $x - 6 = 0$, which gives us $x = 6$. For the second factor, $x + 5 = 0$, which gives us $x = -5$. So, the solutions to the equation $x^2 - x - 30 = 0$ are $x = 6$ and $x = -5$. These are the two points where the parabola crosses the x-axis. This factoring technique is a cornerstone of algebra, and understanding it will boost your problem-solving skills considerably. It's often helpful to list out the factor pairs of the constant term to help you find the right combination.

Let's outline the steps again.

1. Find the Numbers: Identify two numbers that multiply to -30 and add up to -1.
2. Factor: Factor the equation into $(x - 6)(x + 5) = 0$.
3. Solve: Set each factor to zero: $x - 6 = 0$ and $x + 5 = 0$.
4. Isolate x: Solve for x: $x = 6$ and $x = -5$.

Therefore, the solutions are $x = 6$ and $x = -5$. This method is a standard approach to solving quadratic equations where the leading coefficient is 1. The key lies in finding the correct factor pairs that satisfy both multiplication and addition conditions.

Solving $9x^2 - 49 = 0$: Difference of Squares

Moving on, let's tackle $9x^2 - 49 = 0$. This is an example of a difference of squares. The difference of squares is a special factoring pattern that appears when you have two perfect squares separated by a subtraction sign. In this case, $9x^2$ is the square of $3x$, and $49$ is the square of $7$. The formula for factoring a difference of squares is $a^2 - b^2 = (a - b)(a + b)$. Applying this to our equation, we get $(3x - 7)(3x + 7) = 0$. Now, we set each factor equal to zero and solve for x. For the first factor, $3x - 7 = 0$, which gives us $3x = 7$, and thus x = rac{7}{3}. For the second factor, $3x + 7 = 0$, which gives us $3x = -7$, and thus x = -rac{7}{3}. Therefore, the solutions to the equation $9x^2 - 49 = 0$ are x = rac{7}{3} and x = -rac{7}{3}. This is a common pattern to recognize, as it allows for a straightforward factorization. The two solutions are equidistant from zero on the number line, a characteristic of equations that represent symmetrical parabolas.

Here are the steps in detail.

1. Recognize the Pattern: Identify $9x^2 - 49$ as a difference of squares.
2. Factor: Factor the expression into $(3x - 7)(3x + 7) = 0$.
3. Solve: Set each factor to zero: $3x - 7 = 0$ and $3x + 7 = 0$.
4. Isolate x: Solve for x: x = rac{7}{3} and x = -rac{7}{3}.

Thus, the solutions are x = rac{7}{3} and x = -rac{7}{3}. The difference of squares is a powerful factoring technique that simplifies many quadratic equations, making them easier to solve.

Solving $3x^2 + 17x - 6 = 0$: Factoring with a Leading Coefficient

Finally, let's solve $3x^2 + 17x - 6 = 0$. This equation involves factoring a quadratic expression where the leading coefficient (the coefficient of $x^2$) is not equal to $1$. This type of factoring requires a slightly different approach. One method is to use the