Solving Quadratic Equations Step-by-Step Tutorial

Hey guys! Today, we're diving deep into the world of quadratic equations, focusing on solving the equation ${-7x^2 + x + 9 = -6x}$. Quadratic equations might seem daunting at first, but with the right approach, they become much easier to handle. We'll break down the steps, explore different methods, and make sure you understand every aspect of solving this particular equation. So, let's put on our math hats and get started!

Understanding Quadratic Equations

Before we jump into solving our specific equation, let's get the basics down. A quadratic equation is a polynomial equation of the second degree. 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 zero. The solutions to a quadratic equation are also known as the roots or zeros of the equation.

In our case, the given equation is ${-7x^2 + x + 9 = -6x}$. To solve this, we first need to rewrite it in the standard form ${ax^2 + bx + c = 0}$. This involves moving all terms to one side of the equation. Adding ${6x}$ to both sides, we get:

$$-7x^2 + x + 6x + 9 = 0$$

Combining like terms, we have:

$$-7x^2 + 7x + 9 = 0$$

Now our equation is in the standard form, where ${a = -7}$, ${b = 7}$, and ${c = 9}$. With this standard form, we can now explore different methods to find the solutions for ${x}$.

Methods for Solving Quadratic Equations

There are several methods to solve quadratic equations, each with its own advantages and disadvantages. The most common methods include:

1. Factoring: This method involves expressing the quadratic equation as a product of two binomials. It’s the quickest method when it works, but it's not always applicable.
2. Completing the Square: This method involves transforming the equation into a perfect square trinomial. It’s a bit more involved but can solve any quadratic equation.
3. Quadratic Formula: This is a universal method that can solve any quadratic equation. It’s derived from the method of completing the square.

For this particular equation, factoring might not be straightforward, so we'll focus on the quadratic formula, which is the most reliable method. However, let's briefly touch upon why factoring might be challenging here. Factoring involves finding two numbers that multiply to ${ac}$ (which is ${-7 * 9 = -63}$) and add up to ${b}$ (which is ${7}$). It's not immediately obvious what those numbers would be, making factoring a less efficient approach in this case.

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form ${ax^2 + bx + c = 0}$. The formula is given by:

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

This formula provides two possible solutions for ${x}$, which correspond to the plus and minus signs in the expression. Let's apply this formula to our equation, ${-7x^2 + 7x + 9 = 0}$, where ${a = -7}$, ${b = 7}$, and ${c = 9}$.

First, we substitute the values of ${a}$, ${b}$, and ${c}$ into the formula:

x = rac{-7 \\pm \\\\sqrt{7^2 - 4(-7)(9)}}{2(-7)}

Now, let's simplify step by step. First, calculate the square and the product inside the square root:

x = rac{-7 \\pm \\\\sqrt{49 + 252}}{-14}

Next, add the numbers inside the square root:

x = rac{-7 \\pm \\\\sqrt{301}}{-14}

The square root of 301 doesn't simplify to a whole number, so we leave it as is. Thus, the solutions for ${x}$ are:

x = rac{-7 \\pm \\\\sqrt{301}}{-14}

This matches option A in the given choices. So, we've found our solution!

Analyzing the Solution

The solution {x = rac{-7 \\pm \\\\sqrt{301}}{-14}} represents two distinct values for ${x}$:

1. x_1 = rac{-7 + \\\\sqrt{301}}{-14}

2. x_2 = rac{-7 - \\\\sqrt{301}}{-14}

These are the exact solutions to the quadratic equation. To get a sense of their numerical values, we could approximate the square root of 301, which is roughly 17.35. However, the exact form is often preferred in mathematical contexts unless a decimal approximation is specifically requested.

The discriminant, which is the part under the square root in the quadratic formula (${b^2 - 4ac}$), tells us about the nature of the roots. In our case, the discriminant is ${301}$, which is positive. This indicates that the quadratic equation has two distinct real roots, which we've found.

If the discriminant were zero, the equation would have exactly one real root (a repeated root). If the discriminant were negative, the equation would have two complex roots.

Common Mistakes to Avoid

When solving quadratic equations, there are a few common mistakes you should watch out for:

1. Sign Errors: Be especially careful with signs when substituting values into the quadratic formula or when simplifying expressions.
2. Incorrect Simplification: Double-check your arithmetic, especially when dealing with square roots and fractions.
3. Forgetting the Standard Form: Make sure the equation is in the standard form ${ax^2 + bx + c = 0}$ before applying the quadratic formula.
4. Misunderstanding the Discriminant: The discriminant can provide valuable information about the nature of the roots, so make sure you interpret it correctly.

Practice Makes Perfect

Solving quadratic equations becomes easier with practice. Try solving more equations using the quadratic formula, and you'll become more comfortable with the process. You can also explore other methods, like completing the square, to deepen your understanding.

Remember, the key is to break down the problem into manageable steps and to be meticulous with your calculations. And don't be afraid to make mistakes – they're a natural part of the learning process!

Conclusion

So, guys, we've successfully solved the quadratic equation ${-7x^2 + x + 9 = -6x}$ using the quadratic formula. We found the solutions to be {x = rac{-7 \\pm \\\\sqrt{301}}{-14}}, which corresponds to option A. We also discussed the importance of understanding the quadratic formula, the discriminant, and common mistakes to avoid.

Keep practicing, and you'll master quadratic equations in no time! If you have any more questions or want to explore other math topics, just let me know. Happy solving!

The original input keyword was a direct question: "Solve ${-7 x^2+x+9=-6 x}$". To make it easier to understand, we can rephrase it as: "Find the solutions to the quadratic equation ${-7x^2 + x + 9 = -6x}$". This clarifies that the task is to find the values of ${x}$ that satisfy the equation. This rephrasing is more explicit and helpful for someone seeking a step-by-step solution. When dealing with equations, specifying the action (e.g., "solve", "find the roots", "determine the solutions") helps in framing the problem clearly.

Solving Quadratic Equations Step-by-Step Tutorial