Factor Completely.${3u^2 - 42u + 72}$

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. In this article, we will focus on factoring the quadratic expression 3u2−42u+723u^2 - 42u + 72 completely. We will break down the process into manageable steps and provide a clear explanation of each step.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form ax2+bx+cax^2 + bx + c, where aa, bb, and cc are constants, and xx is the variable. In our case, the quadratic expression is 3u2−42u+723u^2 - 42u + 72, where a=3a = 3, b=−42b = -42, and c=72c = 72.

Factoring Quadratic Expressions

Factoring a quadratic expression involves finding two binomials whose product is equal to the original expression. In other words, we need to find two binomials (x+y)(x + y) and (x−y)(x - y) such that their product is equal to the quadratic expression.

Step 1: Find the Greatest Common Factor (GCF)

The first step in factoring a quadratic expression is to find the greatest common factor (GCF) of the three terms. The GCF is the largest factor that divides all three terms without leaving a remainder. In our case, the GCF of 3u23u^2, −42u-42u, and 7272 is 33.

GCF(3u^2, -42u, 72) = 3

Step 2: Factor Out the GCF

Once we have found the GCF, we can factor it out of each term. This involves dividing each term by the GCF and writing the result as a product of the GCF and a new expression.

3u^2 - 42u + 72 = 3(u^2 - 14u + 24)

Step 3: Factor the Quadratic Expression

Now that we have factored out the GCF, we can focus on factoring the quadratic expression inside the parentheses. This involves finding two binomials whose product is equal to the quadratic expression.

u^2 - 14u + 24 = (u - 6)(u - 4)

Step 4: Write the Final Factored Form

The final step is to write the factored form of the quadratic expression. This involves combining the GCF and the factored form of the quadratic expression.

3u^2 - 42u + 72 = 3(u - 6)(u - 4)

Conclusion

Factoring quadratic expressions is an essential skill in algebra that helps us simplify complex equations and solve problems more efficiently. By following the steps outlined in this article, we can factor the quadratic expression 3u2−42u+723u^2 - 42u + 72 completely. Remember to find the GCF, factor it out, and then focus on factoring the quadratic expression inside the parentheses.

Example Problems

Problem 1

Factor the quadratic expression 2x2+16x+302x^2 + 16x + 30 completely.

Solution

2x^2 + 16x + 30 = 2(x^2 + 8x + 15) = 2(x + 3)(x + 5)

Problem 2

Factor the quadratic expression 4y2−20y+244y^2 - 20y + 24 completely.

Solution

4y^2 - 20y + 24 = 4(y^2 - 5y + 6) = 4(y - 2)(y - 3)

Tips and Tricks

  • Always start by finding the GCF of the three terms.
  • Factor out the GCF and then focus on factoring the quadratic expression inside the parentheses.
  • Use the distributive property to check your work and ensure that the factored form is correct.

Common Mistakes

  • Failing to find the GCF of the three terms.
  • Factoring out the wrong term or factoring out a term that is not a factor.
  • Not checking the work using the distributive property.

Real-World Applications

Factoring quadratic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, factoring quadratic expressions can be used to model the motion of objects under the influence of gravity, to design electrical circuits, and to analyze economic systems.

Conclusion

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. By following the steps outlined in this article, we can factor the quadratic expression 3u2−42u+723u^2 - 42u + 72 completely. Remember to find the GCF, factor it out, and then focus on factoring the quadratic expression inside the parentheses. With practice and patience, you will become proficient in factoring quadratic expressions and be able to apply this skill to a wide range of real-world problems.

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Introduction

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. In our previous article, we provided a step-by-step guide on how to factor the quadratic expression 3u2−42u+723u^2 - 42u + 72 completely. In this article, we will answer some of the most frequently asked questions about factoring quadratic expressions.

Q&A

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

A: The greatest common factor (GCF) of a quadratic expression is the largest factor that divides all three terms without leaving a remainder.

Q: How do I find the GCF of a quadratic expression?

A: To find the GCF of a quadratic expression, you can list the factors of each term and find the greatest common factor among them.

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

A: Factoring a quadratic expression involves finding two binomials whose product is equal to the original expression, while simplifying a quadratic expression involves combining like terms to reduce the expression to its simplest form.

Q: Can I factor a quadratic expression that has no real roots?

A: Yes, you can factor a quadratic expression that has no real roots. In this case, the factored form will be in the form of complex numbers.

Q: How do I check my work when factoring a quadratic expression?

A: To check your work, you can use the distributive property to multiply the factored form and ensure that it is equal to the original expression.

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

A: Some common mistakes to avoid when factoring quadratic expressions include failing to find the GCF, factoring out the wrong term, and not checking the work using the distributive property.

Q: Can I use factoring to solve quadratic equations?

A: Yes, you can use factoring to solve quadratic equations. By setting the factored form equal to zero, you can solve for the variable.

Q: How do I factor a quadratic expression with a negative leading coefficient?

A: To factor a quadratic expression with a negative leading coefficient, you can multiply the entire expression by -1 to make the leading coefficient positive, and then factor the expression as usual.

Q: Can I factor a quadratic expression with a variable in the denominator?

A: No, you cannot factor a quadratic expression with a variable in the denominator. In this case, you will need to use other methods, such as the quadratic formula, to solve the equation.

Conclusion

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. By understanding the GCF, factoring, and simplifying quadratic expressions, you can apply this skill to a wide range of real-world problems. Remember to check your work using the distributive property and avoid common mistakes such as failing to find the GCF and factoring out the wrong term.

Example Problems

Problem 1

Factor the quadratic expression 2x2+16x+302x^2 + 16x + 30 completely.

Solution

2x^2 + 16x + 30 = 2(x^2 + 8x + 15) = 2(x + 3)(x + 5)

Problem 2

Factor the quadratic expression 4y2−20y+244y^2 - 20y + 24 completely.

Solution

4y^2 - 20y + 24 = 4(y^2 - 5y + 6) = 4(y - 2)(y - 3)

Tips and Tricks

  • Always start by finding the GCF of the three terms.
  • Factor out the GCF and then focus on factoring the quadratic expression inside the parentheses.
  • Use the distributive property to check your work and ensure that the factored form is correct.

Common Mistakes

  • Failing to find the GCF of the three terms.
  • Factoring out the wrong term or factoring out a term that is not a factor.
  • Not checking the work using the distributive property.

Real-World Applications

Factoring quadratic expressions has numerous real-world applications in fields such as physics, engineering, and economics. For example, factoring quadratic expressions can be used to model the motion of objects under the influence of gravity, to design electrical circuits, and to analyze economic systems.

Conclusion

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. By understanding the GCF, factoring, and simplifying quadratic expressions, you can apply this skill to a wide range of real-world problems. Remember to check your work using the distributive property and avoid common mistakes such as failing to find the GCF and factoring out the wrong term.