Solving The Quadratic Equation: 10 = -4x + 3x^2

Hey guys! Let's dive into solving this quadratic equation. Quadratic equations might seem intimidating at first, but once you break them down, they're totally manageable. In this article, we'll walk through the steps to solve the equation $10 = -4x + 3x^2$ and find the correct solution from the given options. We’ll cover everything from rearranging the equation into standard form to applying the quadratic formula. So, grab your calculators and let's get started!

Understanding Quadratic Equations

Before we jump into the specifics of our equation, let’s quickly recap what a quadratic equation is. A quadratic equation is a polynomial equation of the second degree. This means the highest power of the variable (in our case, x) is 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. The solutions to a quadratic equation are also known as its roots or zeros. These are the values of x that make the equation true. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula. Each method has its own advantages, but the quadratic formula is often the most reliable, especially for equations that are difficult to factor.

When dealing with quadratic equations, you'll often encounter situations where you need to rearrange the terms to fit the standard form. This rearrangement is crucial because it allows you to easily identify the coefficients a, b, and c, which are essential for solving the equation using methods like the quadratic formula. Additionally, understanding the structure of a quadratic equation helps in visualizing its graph, which is a parabola. The roots of the equation correspond to the points where the parabola intersects the x-axis. This visual representation can provide valuable insights into the nature of the solutions and the behavior of the quadratic function. Therefore, grasping the fundamentals of quadratic equations, including their standard form and graphical representation, is key to mastering algebraic problem-solving.

The Importance of the Standard Form

Getting the equation into the standard form $ax^2 + bx + c = 0$ is super important because it sets us up perfectly to use methods like the quadratic formula. This form helps us clearly identify the coefficients a, b, and c, which are the keys to unlocking the solution. Plus, when the equation is in standard form, it's much easier to analyze and understand its properties. For instance, you can quickly determine the discriminant ($b^2 - 4ac$), which tells you about the nature of the roots (whether they are real, distinct, or complex). Furthermore, the standard form makes it straightforward to complete the square, another powerful method for solving quadratic equations. So, remember, the standard form isn't just a formality; it's your best friend when tackling quadratic equations!

Step-by-Step Solution

Now, let's tackle the equation $10 = -4x + 3x^2$ step by step. We’ll follow a clear process to make sure we get the correct solution. Let's break it down:

1. Rearrange the Equation

Our first task is to get the equation into the standard form $ax^2 + bx + c = 0$. To do this, we need to move all terms to one side of the equation, leaving zero on the other side. The given equation is:

$10 = -4x + 3x^2$

Subtract 10 from both sides to set the equation to zero:

$0 = 3x^2 - 4x - 10$

Now, we have the equation in the standard form, where:

• a = 3
• b = -4
• c = -10

2. Apply the Quadratic Formula

Since this equation doesn't factor easily, the best approach is to use the quadratic formula. The quadratic formula is a powerful tool that gives us the solutions for any quadratic equation in standard form. The formula is:

$x = \frac{-b rac{+}{-}

\sqrt{b^2 - 4ac}}{2a}$

Plug in the values of a, b, and c that we identified earlier:

$x = \frac{-(-4) rac{+}{-}

\sqrt{(-4)^2 - 4(3)(-10)}}{2(3)}$

3. Simplify the Expression

Next, we need to simplify the expression inside the square root and the rest of the formula. Let's break it down:

First, simplify the terms inside the square root:

$(-4)^2 = 16$

$4(3)(-10) = -120$

So, the expression inside the square root becomes:

$16 - (-120) = 16 + 120 = 136$

Now, plug this back into the formula:

$x = \frac{4 rac{+}{-}

\sqrt{136}}{6}$

4. Further Simplification

We can simplify the square root of 136 by finding its prime factors. 136 can be factored as $4 * 34$, so $\sqrt{136}$ becomes $\sqrt{4 * 34}$. Since $\sqrt{4} = 2$, we can rewrite $\sqrt{136}$ as $2\sqrt{34}$.

Now, our equation looks like this:

x = \frac{4 rac{+}{-} 2\sqrt{34}}{6}

We can simplify this further by dividing each term in the numerator by the denominator (6):

$x = \frac{4}{6} rac{+}{-}

\frac{2\sqrt{34}}{6}$

Reduce the fractions:

$x = \frac{2}{3} rac{+}{-}

\frac{\sqrt{34}}{3}$

Combine the terms:

x = \frac{2 rac{+}{-} \sqrt{34}}{3}

5. Compare with the Options

Now, we have the solutions for x:

$x = \frac{2 + \sqrt{34}}{3}$ and $x = \frac{2 - \sqrt{34}}{3}$.

Let’s compare this with the given options:

(A) $x=1,-\frac{1}{2}$ (B) x=\frac{-1 rac{+}{-} \sqrt{10}}{2} (C) x=\frac{-2 rac{+}{-} \sqrt{34}}{-3} (D) x=\frac{-3 rac{+}{-} \sqrt{17}}{2}

Our solution x = \frac{2 rac{+}{-} \sqrt{34}}{3} matches option (C) if we multiply both the numerator and the denominator by -1:

x = \frac{-1(2 rac{+}{-} \sqrt{34})}{-1(3)} = \frac{-2 rac{+}{-} \sqrt{34}}{-3}

Choosing the Correct Answer

After carefully working through the steps and simplifying our solution, we can confidently choose the correct answer. So, the correct answer is:

(C) x=\frac{-2 rac{+}{-} \sqrt{34}}{-3}

Tips for Solving Quadratic Equations

To wrap things up, let's go over some handy tips that can help you nail solving quadratic equations every time. These tips can save you time and reduce the chances of making errors.

1. Always Check for Standard Form

Before you do anything else, make sure your equation is in the standard form ($ax^2 + bx + c = 0$). As we discussed earlier, this form makes it much easier to identify the coefficients a, b, and c, which are crucial for applying the quadratic formula or completing the square. If the equation isn't in standard form, rearrange it first. Trust me, this simple step can prevent a lot of headaches down the road!

2. Try Factoring First

Factoring is often the quickest way to solve a quadratic equation, if it's possible. Look for two numbers that multiply to c and add up to b. If you can find these numbers, you can easily factor the quadratic and solve for x. Factoring is especially useful for equations with integer solutions. However, don't spend too much time trying to factor if it's not immediately obvious; after a few attempts, it might be more efficient to move on to the quadratic formula.

3. Master the Quadratic Formula

The quadratic formula is your best friend when factoring doesn't work. Memorize it, understand it, and practice using it. The formula is:

$x = \frac{-b rac{+}{-}

\sqrt{b^2 - 4ac}}{2a}$

Make sure you correctly identify a, b, and c, and plug them into the formula carefully. Pay close attention to the signs (positive and negative) to avoid common mistakes. The quadratic formula will always give you the solutions, whether they are real or complex.

4. Simplify Radicals

After applying the quadratic formula, you'll often end up with a radical (square root) in your solution. Make sure to simplify the radical as much as possible. Look for perfect square factors inside the square root and take them out. For example, $\sqrt{12}$ can be simplified to $2\sqrt{3}$ because 12 = 4 * 3 and $\sqrt{4} = 2$. Simplifying radicals not only gives you the most accurate answer but also makes it easier to compare your solution with the given options.

5. Double-Check Your Work

It's always a good idea to double-check your work, especially in math. After you've found your solutions, plug them back into the original equation to make sure they work. This can help you catch any mistakes you might have made along the way. Also, make sure you've answered the question completely. Sometimes, there might be multiple parts to the question, or you might need to provide your answer in a specific format.

6. Practice, Practice, Practice!

Like any skill, solving quadratic equations gets easier with practice. Work through lots of examples, and don't be afraid to make mistakes. Mistakes are a great way to learn! The more you practice, the more comfortable you'll become with the different methods and the more confident you'll feel when tackling these types of problems. You can find practice problems in textbooks, online resources, or from your teacher.

By following these tips, you'll be well-equipped to solve any quadratic equation that comes your way. Keep practicing, stay patient, and you'll become a quadratic equation pro in no time!

Conclusion

So, there you have it! We successfully solved the quadratic equation $10 = -4x + 3x^2$ by rearranging it into standard form, applying the quadratic formula, and simplifying the result. Remember, the key to mastering quadratic equations is practice and understanding the fundamental concepts. Keep these steps in mind, and you’ll be solving quadratic equations like a pro in no time. Keep up the great work, and happy solving!