Simplify The Expression:${ 9y^2 - 12y + 4 }$

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Introduction

In this article, we will simplify the given expression: 9y^2 - 12y + 4. This involves factoring the quadratic expression, which is a fundamental concept in algebra. Factoring a quadratic expression allows us to rewrite it in a more simplified form, making it easier to solve equations and perform other mathematical operations.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means the highest power of the variable (in this case, y) is two. The general form of a quadratic expression is:

ax^2 + bx + c

where a, b, and c are constants, and x is the variable. In our given expression, 9y^2 - 12y + 4, a = 9, b = -12, and c = 4.

Factoring the Quadratic Expression

To factor the quadratic expression, we need to find two numbers whose product is equal to the product of the coefficient of the squared term (a) and the constant term (c), and whose sum is equal to the coefficient of the linear term (b). In this case, we need to find two numbers whose product is equal to 9 × 4 = 36, and whose sum is equal to -12.

After some trial and error, we find that the two numbers are -6 and -6, since (-6) × (-6) = 36 and (-6) + (-6) = -12.

Writing the Factored Form

Now that we have found the two numbers, we can write the factored form of the quadratic expression:

9y^2 - 12y + 4 = 3y^2 - 6y + 2y - 2

We can then factor out the common terms:

9y^2 - 12y + 4 = 3y(y - 2) + 2(y - 2)

Simplifying the Factored Form

We can see that both terms in the factored form have a common factor of (y - 2). We can factor this out to get:

9y^2 - 12y + 4 = (3y + 2)(y - 2)

This is the simplified form of the given expression.

Conclusion

In this article, we simplified the quadratic expression 9y^2 - 12y + 4 by factoring it. We found the two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the linear term. We then wrote the factored form of the expression and simplified it further by factoring out the common terms.

Real-World Applications

Factoring quadratic expressions has many real-world applications, such as:

  • Solving equations: Factoring quadratic expressions allows us to solve equations of the form ax^2 + bx + c = 0.
  • Graphing functions: Factoring quadratic expressions allows us to graph functions of the form f(x) = ax^2 + bx + c.
  • Optimization: Factoring quadratic expressions allows us to optimize functions of the form f(x) = ax^2 + bx + c.

Common Mistakes

When factoring quadratic expressions, it's common to make mistakes such as:

  • Not finding the correct two numbers: Make sure to find the correct two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the linear term.
  • Not factoring out the common terms: Make sure to factor out the common terms in the factored form of the expression.

Tips and Tricks

When factoring quadratic expressions, here are some tips and tricks to keep in mind:

  • Use the product and sum method: Use the product and sum method to find the two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the linear term.
  • Check for common factors: Check for common factors in the factored form of the expression and factor them out.
  • Use the distributive property: Use the distributive property to simplify the factored form of the expression.

Conclusion

Introduction

In our previous article, we simplified the quadratic expression 9y^2 - 12y + 4 by factoring it. In this article, we will answer some common questions related to factoring quadratic expressions.

Q&A

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

A: Factoring a quadratic expression involves rewriting it in a product of two binomials, while simplifying a quadratic expression involves rewriting it in a more compact form.

Q: How do I know if a quadratic expression can be factored?

A: A quadratic expression can be factored if it can be rewritten in a product of two binomials. To determine if a quadratic expression can be factored, try to find two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the linear term.

Q: What is the product and sum method?

A: The product and sum method is a technique used to factor quadratic expressions. It involves finding two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the linear term.

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

A: To factor a quadratic expression with a negative coefficient, simply factor out the negative sign and then factor the remaining expression.

Q: Can a quadratic expression have more than two factors?

A: Yes, a quadratic expression can have more than two factors. However, it is not always possible to factor a quadratic expression into more than two factors.

Q: How do I check if a factored form is correct?

A: To check if a factored form is correct, multiply the two binomials together and simplify the expression. If the result is the original quadratic expression, then the factored form is correct.

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

A: Factoring a quadratic expression involves rewriting it in a product of two binomials, while expanding a quadratic expression involves rewriting it in a sum of terms.

Q: Can a quadratic expression be factored if it has a complex coefficient?

A: Yes, a quadratic expression can be factored if it has a complex coefficient. However, the factored form may involve complex numbers.

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

A: To factor a quadratic expression with a variable coefficient, try to find two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the linear term.

Q: Can a quadratic expression be factored if it has a negative leading coefficient?

A: Yes, a quadratic expression can be factored if it has a negative leading coefficient. However, the factored form may involve negative signs.

Conclusion

In conclusion, factoring quadratic expressions is a fundamental concept in algebra that has many real-world applications. By understanding how to factor quadratic expressions, we can solve equations, graph functions, and optimize functions. Remember to use the product and sum method, check for common factors, and use the distributive property to simplify the factored form of the expression.

Common Mistakes

When factoring quadratic expressions, it's common to make mistakes such as:

  • Not finding the correct two numbers: Make sure to find the correct two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the linear term.
  • Not factoring out the common terms: Make sure to factor out the common terms in the factored form of the expression.
  • Not checking if the factored form is correct: Make sure to multiply the two binomials together and simplify the expression to check if the factored form is correct.

Tips and Tricks

When factoring quadratic expressions, here are some tips and tricks to keep in mind:

  • Use the product and sum method: Use the product and sum method to find the two numbers whose product is equal to the product of the coefficient of the squared term and the constant term, and whose sum is equal to the coefficient of the linear term.
  • Check for common factors: Check for common factors in the factored form of the expression and factor them out.
  • Use the distributive property: Use the distributive property to simplify the factored form of the expression.
  • Use complex numbers: Use complex numbers to factor quadratic expressions with complex coefficients.
  • Use the quadratic formula: Use the quadratic formula to solve quadratic equations.

Conclusion

In conclusion, factoring quadratic expressions is a fundamental concept in algebra that has many real-world applications. By understanding how to factor quadratic expressions, we can solve equations, graph functions, and optimize functions. Remember to use the product and sum method, check for common factors, and use the distributive property to simplify the factored form of the expression.