Solve For \[$ X \$\] In The Equation: $\[ 5x^2 - 2x = 16 \\]

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 quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. We will use the given equation $5x^2 - 2x = 16$ as a case study to demonstrate the steps involved in solving quadratic equations.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The quadratic formula can be used to solve quadratic equations that cannot be factored easily.

Solving the Given Equation

Now, let's apply the quadratic formula to solve the given equation $5x^2 - 2x = 16$. First, we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting 16 from both sides of the equation:

$$5x^2 - 2x - 16 = 0$$

Now, we can identify the coefficients $a$, $b$, and $c$:

$$a = 5, b = -2, c = -16$$

Next, we can plug these values into the quadratic formula:

$$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(5)(-16)}}{2(5)}$$

Simplifying the expression, we get:

$$x = \frac{2 \pm \sqrt{4 + 320}}{10}$$

$$x = \frac{2 \pm \sqrt{324}}{10}$$

$$x = \frac{2 \pm 18}{10}$$

Now, we have two possible solutions:

$$x = \frac{2 + 18}{10} = \frac{20}{10} = 2$$

$$x = \frac{2 - 18}{10} = \frac{-16}{10} = -\frac{8}{5}$$

Therefore, the solutions to the equation $5x^2 - 2x = 16$ are $x = 2$ and $x = -\frac{8}{5}$.

Conclusion

Solving quadratic equations is an essential skill for students and professionals alike. In this article, we used the quadratic formula to solve the equation $5x^2 - 2x = 16$. We identified the coefficients $a$, $b$, and $c$, and plugged them into the quadratic formula to find the solutions. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to solve equations that cannot be factored easily.

Tips and Tricks

• Make sure to identify the coefficients $a$, $b$, and $c$ correctly before plugging them into the quadratic formula.
• Simplify the expression under the square root sign to make it easier to work with.
• Be careful when adding or subtracting fractions to avoid errors.
• Check your solutions by plugging them back into the original equation to ensure that they are true.

Common Mistakes

• Failing to identify the coefficients $a$, $b$, and $c$ correctly.
• Not simplifying the expression under the square root sign.
• Adding or subtracting fractions incorrectly.
• Not checking the solutions by plugging them 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 the behavior of economic systems, such as supply and demand.
• Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Conclusion

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 answer some of the most frequently asked questions about quadratic equations, including how to solve them, common mistakes to avoid, and real-world applications.

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 is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including factoring, the quadratic formula, and graphing. The quadratic formula is a powerful tool for solving quadratic equations, and it is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the coefficients $a$, $b$, and $c$ of the quadratic equation. Then, you can plug these values into the quadratic formula to find the solutions.

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

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

• Failing to identify the coefficients $a$, $b$, and $c$ correctly.
• Not simplifying the expression under the square root sign.
• Adding or subtracting fractions incorrectly.
• Not checking the solutions by plugging them back into the original equation.

Q: What are the real-world applications of quadratic equations?

A: 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 the behavior of economic systems, such as supply and demand.
• Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Q: How do I check my solutions?

A: To check your solutions, you need to plug them back into the original equation to ensure that they are true. This is an important step to avoid errors and ensure that your solutions are correct.

Q: What are some tips for solving quadratic equations?

A: Some tips for solving quadratic equations include:

• Make sure to identify the coefficients $a$, $b$, and $c$ correctly before plugging them into the quadratic formula.
• Simplify the expression under the square root sign to make it easier to work with.
• Be careful when adding or subtracting fractions to avoid errors.
• Check your solutions by plugging them back into the original equation to ensure that they are true.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the quadratic formula, identifying common mistakes to avoid, and exploring real-world applications, you can solve quadratic equations with confidence and accuracy.