Simplify The Expression:${6x^2 - 42x + 60}$

by ADMIN 45 views

Introduction

In this article, we will simplify the given quadratic expression 6x2−42x+606x^2 - 42x + 60. We will use various algebraic techniques to factorize and simplify the expression. The main goal is to rewrite the expression in a simpler form, making it easier to work with.

Understanding the Expression

The given expression is a quadratic expression in the form of ax2+bx+cax^2 + bx + c. Here, a=6a = 6, b=−42b = -42, and c=60c = 60. To simplify the expression, we need to factorize it, which involves finding the greatest common factor (GCF) of the terms and expressing the expression as a product of two binomials.

Step 1: Factorize the Expression

To factorize the expression, we need to find the GCF of the terms. The GCF of 6x26x^2, −42x-42x, and 6060 is 66. We can factor out 66 from each term:

6x2−42x+60=6(x2−7x+10)6x^2 - 42x + 60 = 6(x^2 - 7x + 10)

Step 2: Factorize the Quadratic Expression

Now, we need to factorize the quadratic expression x2−7x+10x^2 - 7x + 10. We can use the factoring method to find two numbers whose product is 1010 and whose sum is −7-7. The numbers are −5-5 and −2-2. We can rewrite the quadratic expression as:

x2−7x+10=(x−5)(x−2)x^2 - 7x + 10 = (x - 5)(x - 2)

Step 3: Simplify the Expression

Now, we can simplify the original expression by substituting the factorized form of the quadratic expression:

6x2−42x+60=6(x2−7x+10)=6(x−5)(x−2)6x^2 - 42x + 60 = 6(x^2 - 7x + 10) = 6(x - 5)(x - 2)

Conclusion

In this article, we simplified the given quadratic expression 6x2−42x+606x^2 - 42x + 60 using algebraic techniques. We factorized the expression by finding the GCF and expressing it as a product of two binomials. The simplified expression is 6(x−5)(x−2)6(x - 5)(x - 2).

Real-World Applications

Simplifying quadratic expressions is an essential skill in mathematics, with numerous real-world applications. For example, in physics, quadratic expressions are used to model the motion of objects under the influence of gravity. In engineering, quadratic expressions are used to design and optimize systems, such as bridges and buildings.

Tips and Tricks

When simplifying quadratic expressions, it's essential to remember the following tips and tricks:

  • Factor out the GCF to simplify the expression.
  • Use the factoring method to factorize quadratic expressions.
  • Check for common factors and simplify the expression accordingly.
  • Use algebraic techniques, such as substitution and elimination, to simplify the expression.

Common Mistakes

When simplifying quadratic expressions, it's easy to make mistakes. Here are some common mistakes to avoid:

  • Failing to factor out the GCF.
  • Not using the factoring method to factorize quadratic expressions.
  • Not checking for common factors and simplifying the expression accordingly.
  • Not using algebraic techniques, such as substitution and elimination, to simplify the expression.

Conclusion

In conclusion, simplifying quadratic expressions is an essential skill in mathematics, with numerous real-world applications. By following the steps outlined in this article, you can simplify quadratic expressions and become proficient in algebra. Remember to factor out the GCF, use the factoring method, and check for common factors to simplify the expression. With practice and patience, you can master the art of simplifying quadratic expressions.

Final Answer

Introduction

In our previous article, we simplified the given quadratic expression 6x2−42x+606x^2 - 42x + 60 using algebraic techniques. In this article, we will answer some frequently asked questions (FAQs) related to simplifying quadratic expressions.

Q: What is the greatest common factor (GCF) of the terms in a quadratic expression?

A: The GCF of the terms in a quadratic expression is the largest number that divides each term without leaving a remainder. In the given expression 6x2−42x+606x^2 - 42x + 60, the GCF is 66.

Q: How do I factorize a quadratic expression?

A: To factorize a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. In the given expression x2−7x+10x^2 - 7x + 10, the numbers are −5-5 and −2-2.

Q: What is the difference between factoring and simplifying a quadratic expression?

A: Factoring a quadratic expression involves expressing it as a product of two binomials, while simplifying a quadratic expression involves rewriting it in a simpler form. In the given expression 6x2−42x+606x^2 - 42x + 60, we factored it as 6(x−5)(x−2)6(x - 5)(x - 2) and then simplified it to 6(x−5)(x−2)6(x - 5)(x - 2).

Q: How do I check if a quadratic expression is factorable?

A: To check if a quadratic expression is factorable, you need to look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term. If such numbers exist, then the expression is factorable.

Q: What are some common mistakes to avoid when simplifying quadratic expressions?

A: Some common mistakes to avoid when simplifying quadratic expressions include:

  • Failing to factor out the GCF.
  • Not using the factoring method to factorize quadratic expressions.
  • Not checking for common factors and simplifying the expression accordingly.
  • Not using algebraic techniques, such as substitution and elimination, to simplify the expression.

Q: How do I use algebraic techniques to simplify quadratic expressions?

A: Algebraic techniques, such as substitution and elimination, can be used to simplify quadratic expressions. For example, you can substitute a variable with an expression to simplify the quadratic expression.

Q: What are some real-world applications of simplifying quadratic expressions?

A: Simplifying quadratic expressions has numerous real-world applications, including:

  • Modeling the motion of objects under the influence of gravity in physics.
  • Designing and optimizing systems, such as bridges and buildings, in engineering.
  • Solving problems in computer science, such as finding the shortest path between two nodes in a graph.

Q: How do I practice simplifying quadratic expressions?

A: To practice simplifying quadratic expressions, you can try the following:

  • Start with simple quadratic expressions and move on to more complex ones.
  • Use online resources, such as math websites and apps, to practice simplifying quadratic expressions.
  • Work on problems from math textbooks and worksheets to practice simplifying quadratic expressions.

Conclusion

In conclusion, simplifying quadratic expressions is an essential skill in mathematics, with numerous real-world applications. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying quadratic expressions. Remember to factor out the GCF, use the factoring method, and check for common factors to simplify the expression.

Final Answer

The final answer is: 6(x−5)(x−2)\boxed{6(x - 5)(x - 2)}