Solve The Equation For $d$:$\[4d^2 + 17d + 10 = -d^2\\]

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $4d^2 + 17d + 10 = -d^2$, to find the value of $d$. We will break down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. In this case, the equation is $4d^2 + 17d + 10 = -d^2$. To solve for $d$, we need to isolate the variable $d$ on one side of the equation.

Step 1: Move all terms to one side

The first step in solving the equation is to move all terms to one side of the equation. This will allow us to set the equation equal to zero and use algebraic manipulations to solve for $d$.

4d^2 + 17d + 10 = -d^2
4d^2 + 17d + 10 + d^2 = 0
5d^2 + 17d + 10 = 0

Step 2: Factor the quadratic expression

The next step is to factor the quadratic expression on the left-hand side of the equation. This will allow us to use the factored form to solve for $d$.

5d^2 + 17d + 10 = (5d + 2)(d + 5) = 0

Step 3: Solve for $d$

Now that we have factored the quadratic expression, we can set each factor equal to zero and solve for $d$.

5d + 2 = 0 \Rightarrow d = -\frac{2}{5}
d + 5 = 0 \Rightarrow d = -5

Conclusion

In this article, we have solved the quadratic equation $4d^2 + 17d + 10 = -d^2$ to find the value of $d$. We used algebraic manipulations and mathematical concepts to arrive at the final answer. The solution involved moving all terms to one side of the equation, factoring the quadratic expression, and solving for $d$. We found two possible values for $d$: $d = -\frac{2}{5}$ and $d = -5$.

Tips and Tricks

• When solving quadratic equations, it's essential to move all terms to one side of the equation to set the equation equal to zero.
• Factoring the quadratic expression can be a powerful tool for solving quadratic equations.
• When solving for $d$, make sure to check for extraneous solutions by plugging the solution back into the original equation.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Final Thoughts

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task. In this article, we will address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this complex topic.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two. It is typically written in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use one of the following methods:

• Factoring: If the quadratic expression can be factored into the product of two binomials, you can set each factor equal to zero and solve for x.
• Quadratic formula: If the quadratic expression cannot be factored, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.
• Graphing: You can also graph the quadratic function and find the x-intercepts, which represent the solutions to the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a, where a, b, and c are the coefficients of the quadratic equation.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula: b^2 - 4ac. It determines the nature of the solutions to the quadratic equation.

• If the discriminant is positive, the equation has two distinct real solutions.
• If the discriminant is zero, the equation has one real solution.
• If the discriminant is negative, the equation has no real solutions.

Q: How do I determine the nature of the solutions?

A: To determine the nature of the solutions, you can use the discriminant. If the discriminant is:

• Positive, the equation has two distinct real solutions.
• Zero, the equation has one real solution.
• Negative, the equation has no real solutions.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable, while a linear equation does not.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to find the solutions.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not moving all terms to one side of the equation.
• Not factoring the quadratic expression correctly.
• Not using the correct formula or method.
• Not checking for extraneous solutions.

Conclusion

Quadratic equations can be a challenging topic, but with practice and patience, you can become proficient in solving them. By understanding the concepts and formulas outlined in this article, you can tackle more complex mathematical problems and develop a deeper appreciation for the beauty of mathematics.