Find P And Q: Solve $4x^2 + 12x = 4(x+p)^2 - Q$

Let's dive into solving this equation to find the values of $p$ and $q$ that make the given expressions equal. This involves algebraic manipulation and a bit of pattern recognition. We aim to rewrite the right-hand side of the equation in a form that allows us to directly compare coefficients with the left-hand side. Understanding these quadratic equations and their transformations is key to solving a variety of math problems, especially in algebra and calculus.

Expanding and Simplifying the Equation

To begin, we need to expand the right side of the equation: $4(x+p)^2 - q$. This involves squaring the binomial $(x+p)$ and then distributing the 4, finally subtracting $q$. Let's go through it step by step:

1. Expand $(x+p)^2$: $(x+p)^2 = x^2 + 2px + p^2$
2. Multiply by 4: $4(x^2 + 2px + p^2) = 4x^2 + 8px + 4p^2$
3. Subtract $q$: $4x^2 + 8px + 4p^2 - q$

Now our equation looks like this: $4x^2 + 12x = 4x^2 + 8px + 4p^2 - q$. The goal here is to make the expressions on both sides identical by choosing appropriate values for $p$ and $q$. By carefully expanding and simplifying, we've set the stage for comparing the coefficients of the corresponding terms. This is a standard algebraic technique that allows us to solve for unknown variables when we have equivalent polynomial expressions. Spotting these opportunities and applying the correct expansion is a critical skill.

Comparing Coefficients

Now that we have the equation $4x^2 + 12x = 4x^2 + 8px + 4p^2 - q$, we can compare the coefficients of the corresponding terms on both sides. This means comparing the coefficients of $x^2$, $x$, and the constant terms.

• Coefficient of $x^2$: On both sides, the coefficient of $x^2$ is 4. This doesn't give us any new information but confirms our expansion is correct.
• Coefficient of $x$: On the left side, the coefficient of $x$ is 12. On the right side, it's $8p$. Therefore, we have the equation $12 = 8p$. Solving for $p$, we get $p = \frac{12}{8} = \frac{3}{2}$. This is a crucial step as it directly gives us the value of $p$.
• Constant term: On the left side, the constant term is 0. On the right side, it's $4p^2 - q$. Therefore, we have the equation $0 = 4p^2 - q$. Now we can substitute the value of $p$ we just found to solve for $q$. Substituting $p = \frac{3}{2}$, we get $0 = 4(\frac{3}{2})^2 - q$. Simplifying, $0 = 4(\frac{9}{4}) - q$, which gives $0 = 9 - q$. Therefore, $q = 9$. This final step completes our solution, providing the value for $q$ that satisfies the original equation.

Solution

We have found that $p = \frac{3}{2}$ and $q = 9$. Therefore, the values of $p$ and $q$ that satisfy the equation $4x^2 + 12x = 4(x+p)^2 - q$ are $p = \frac{3}{2}$ and $q = 9$. These values ensure that the quadratic expression on the left is perfectly represented by the transformed expression on the right. To double-check our answer, let's substitute these values back into the original equation and see if it holds true.

• Substitute $p = \frac{3}{2}$ and $q = 9$ into $4(x+p)^2 - q$: $4(x + \frac{3}{2})^2 - 9$
• Expand: $4(x^2 + 3x + \frac{9}{4}) - 9 = 4x^2 + 12x + 9 - 9 = 4x^2 + 12x$

As you can see, the result matches the left side of the original equation, $4x^2 + 12x$. Therefore, we can confidently conclude that our solution is correct. Mastering these techniques is incredibly valuable for solving more complex algebraic problems.

Verification

To verify our solution, we substitute $p = \frac{3}{2}$ and $q = 9$ back into the original equation:

$4x^2 + 12x = 4(x + \frac{3}{2})^2 - 9$

Expanding the right side:

$4(x + \frac{3}{2})^2 - 9 = 4(x^2 + 3x + \frac{9}{4}) - 9 = 4x^2 + 12x + 9 - 9 = 4x^2 + 12x$

Since the left side equals the right side, our solution is correct.

Alternative Approach: Completing the Square

Another way to solve this problem is by completing the square. This method directly transforms the quadratic expression into the desired form. Let's start with the given expression $4x^2 + 12x$.

1. Factor out the coefficient of $x^2$: $4(x^2 + 3x)$

2. Complete the square inside the parenthesis: To complete the square for $x^2 + 3x$, we need to add and subtract $(\frac{3}{2})^2 = \frac{9}{4}$ inside the parenthesis.

   $4(x^2 + 3x + \frac{9}{4} - \frac{9}{4}) = 4((x + \frac{3}{2})^2 - \frac{9}{4})$

3. Distribute the 4: $4(x + \frac{3}{2})^2 - 4(\frac{9}{4}) = 4(x + \frac{3}{2})^2 - 9$

Comparing this with $4(x+p)^2 - q$, we can directly see that $p = \frac{3}{2}$ and $q = 9$. This method provides a more direct route to the solution by leveraging the technique of completing the square, which is a fundamental concept in algebra. The advantage of this approach is that it minimizes the need for coefficient comparison, directly leading to the values of $p$ and $q$.

Importance of Understanding Quadratic Equations

Understanding quadratic equations is fundamental in mathematics and has numerous applications in various fields, including physics, engineering, and computer science. The ability to manipulate and solve quadratic equations is a crucial skill for any student pursuing STEM fields. Quadratic equations appear in diverse contexts, such as projectile motion in physics, circuit analysis in electrical engineering, and optimization problems in computer science. Mastering these concepts not only helps in solving mathematical problems but also enhances problem-solving skills applicable in a wide range of real-world scenarios. Understanding the different forms of quadratic equations and how to transform them is essential for success in higher-level mathematics and related disciplines. Different techniques for solving such equations, such as factoring, completing the square, and using the quadratic formula, each provide unique insights and approaches that can be adapted to various problem scenarios.

Real-World Applications

Quadratic equations aren't just abstract math; they pop up in real-world scenarios more often than you might think! For example, consider the trajectory of a ball thrown into the air. The path it follows can be modeled using a quadratic equation, allowing us to predict how far it will travel and how high it will go. In engineering, quadratic equations are used to design bridges, buildings, and other structures, ensuring they can withstand various forces and stresses. In computer graphics, quadratic equations help create smooth curves and surfaces, making images more realistic and visually appealing. Even in finance, quadratic equations can be used to model investment growth and calculate optimal strategies for maximizing returns. So, next time you see a curved line or a parabolic shape, remember that quadratic equations are working behind the scenes to make it all possible. Grasping the essence of these equations unlocks a deeper understanding of the world around us and equips us with the tools to solve a wide range of practical problems.

By understanding how to manipulate quadratic expressions and solve quadratic equations, you gain valuable tools for tackling a variety of mathematical and real-world problems. Whether it's finding the vertex of a parabola, determining the roots of a polynomial, or optimizing a physical system, the knowledge of quadratic equations will serve you well. The ability to transform equations and compare coefficients is a fundamental skill that will prove invaluable throughout your mathematical journey. Keep practicing these techniques to strengthen your understanding and build confidence in your problem-solving abilities.

In conclusion, by expanding the equation, comparing coefficients, and verifying our solution, we found that $p = \frac{3}{2}$ and $q = 9$. These values satisfy the given equation $4x^2 + 12x = 4(x+p)^2 - q$.

For further learning on quadratic equations, you can visit Khan Academy.