Not Distributing the Numbers Outside the Parentheses

Oct 13, 2025 by talynt.org 79 views

Introduction to Solving Quadratic Equations

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to solve them to progress in various mathematical disciplines. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on solving a specific quadratic equation, and we will use the given equation as an example.

The Given Equation

The given equation is $${ 5(x^2 - 10) = 2(2x^2 + 7) }$$. Our goal is to solve for the variable $x$ . To do this, we need to isolate the variable $x$ on one side of the equation.

Step 1: Distribute the Numbers Outside the Parentheses

To start solving the equation, we need to distribute the numbers outside the parentheses. This means we need to multiply the numbers outside the parentheses by each term inside the parentheses.

{ 5(x^2 - 10) = 2(2x^2 + 7) }

{ 5x^2 - 50 = 4x^2 + 14 }

Step 2: Move All Terms to One Side of the Equation

Now that we have distributed the numbers outside the parentheses, we need to move all terms to one side of the equation. This will help us isolate the variable $x$ .

{ 5x^2 - 50 = 4x^2 + 14 }

{ 5x^2 - 4x^2 = 14 + 50 }

Step 3: Combine Like Terms

Now that we have moved all terms to one side of the equation, we need to combine like terms. This means we need to combine the terms with the same variable and exponent.

{ 5x^2 - 4x^2 = 14 + 50 }

{ x^2 = 64 }

Step 4: Take the Square Root of Both Sides

Now that we have combined like terms, we need to take the square root of both sides of the equation. This will help us solve for the variable $x$ .

{ x^2 = 64 }

{ x = \pm \sqrt{64} }

Step 5: Simplify the Square Root

Now that we have taken the square root of both sides of the equation, we need to simplify the square root. This means we need to find the value of the square root.

{ x = \pm \sqrt{64} }

{ x = \pm 8 }

Conclusion

In this article, we have solved a quadratic equation using the given equation as an example. We have used the steps of distributing the numbers outside the parentheses, moving all terms to one side of the equation, combining like terms, taking the square root of both sides, and simplifying the square root to solve for the variable $x$ . The final solution is $x = \pm 8$ .

Final Answer

The final answer is $\boxed{x = \pm 8}$ .

Applications of Solving Quadratic Equations

Solving quadratic equations has numerous applications in various fields, including physics, engineering, and economics. Some of the applications of solving quadratic equations include:

Projectile Motion: Quadratic equations are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
Signal Processing: Quadratic equations are used in signal processing to filter signals and remove noise.
Machine Learning: Quadratic equations are used in machine learning to optimize models and improve their performance.

Tips and Tricks for Solving Quadratic Equations

Here are some tips and tricks for solving quadratic equations:

Use the Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Check Your Work: It is essential to check your work when solving quadratic equations. This will help you ensure that your solution is correct.
Use a Calculator: A calculator can be a useful tool for solving quadratic equations. It can help you simplify the equation and find the solution.
Practice, Practice, Practice: Solving quadratic equations requires practice. The more you practice, the better you will become at solving them.

Common Mistakes to Avoid When Solving Quadratic Equations

Here are some common mistakes to avoid when solving quadratic equations:

Not Distributing the Numbers Outside the Parentheses: Failing to distribute the numbers outside the parentheses can lead to incorrect solutions.
Not Moving All Terms to One Side of the Equation: Failing to move all terms to one side of the equation can lead to incorrect solutions.
Not Combining Like Terms: Failing to combine like terms can lead to incorrect solutions.
Not Taking the Square Root of Both Sides: Failing to take the square root of both sides of the equation can lead to incorrect solutions.

Conclusion

In conclusion, solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to solve them to progress in various mathematical disciplines. In this article, we have solved a quadratic equation using the given equation as an example. We have used the steps of distributing the numbers outside the parentheses, moving all terms to one side of the equation, combining like terms, taking the square root of both sides, and simplifying the square root to solve for the variable $x$ . The final solution is $x = \pm 8$ .

Introduction to Solving Quadratic Equations

Q&A: Solving Quadratic Equations

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.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you need to follow the steps of distributing the numbers outside the parentheses, moving all terms to one side of the equation, combining like terms, taking the square root of both sides, and simplifying the square root.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$ , $b$ , and $c$ in the quadratic equation, and then plug them into the formula.

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

A: Some common mistakes to avoid when solving quadratic equations include not distributing the numbers outside the parentheses, not moving all terms to one side of the equation, not combining like terms, and not taking the square root of both sides.

Q: How do I check my work when solving quadratic equations?

A: To check your work when solving quadratic equations, you need to plug your solution back into the original equation and verify that it is true.

Q: What are some applications of solving quadratic equations?

A: Some applications of solving quadratic equations include projectile motion, optimization, signal processing, and machine learning.

Q: How do I practice solving quadratic equations?

A: To practice solving quadratic equations, you can try solving different types of quadratic equations, such as those with real and complex roots.