Simplify The Expression:$x^2 + 17x + 60$

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. It involves rewriting an expression in a more compact and manageable form, often by combining like terms or factoring out common factors. In this article, we will focus on simplifying the quadratic expression $x^2 + 17x + 60$.

What is a Quadratic Expression?

A quadratic expression is a polynomial expression of degree two, which means it has a highest power of two. It is typically written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic expressions can be simplified using various techniques, including factoring, completing the square, and using the quadratic formula.

Factoring Quadratic Expressions

One of the most common methods for simplifying quadratic expressions is factoring. Factoring involves expressing a quadratic expression as a product of two binomials. To factor a quadratic expression, we need to find two numbers whose product is equal to the constant term ($c$) and whose sum is equal to the coefficient of the linear term ($b$).

Step 1: Find the Factors of the Constant Term

The constant term in the given expression is 60. We need to find two numbers whose product is equal to 60. The factors of 60 are:

• 1 and 60
• 2 and 30
• 3 and 20
• 4 and 15
• 5 and 12
• 6 and 10

Step 2: Find the Sum of the Factors

We need to find the sum of the factors that we found in Step 1. The sum of the factors is:

• 1 + 60 = 61
• 2 + 30 = 32
• 3 + 20 = 23
• 4 + 15 = 19
• 5 + 12 = 17
• 6 + 10 = 16

Step 3: Write the Factored Form

Now that we have found the factors and their sum, we can write the factored form of the quadratic expression. The factored form is:

$(x + 5)(x + 12)$

Simplifying the Expression

Now that we have factored the quadratic expression, we can simplify it by multiplying the two binomials. To multiply the binomials, we need to follow the distributive property, which states that:

$a(b + c) = ab + ac$

Using the distributive property, we can multiply the two binomials as follows:

$(x + 5)(x + 12) = x(x + 12) + 5(x + 12)$

$= x^2 + 12x + 5x + 60$

$= x^2 + 17x + 60$

Conclusion

In this article, we simplified the quadratic expression $x^2 + 17x + 60$ by factoring it into the form $(x + 5)(x + 12)$. We then multiplied the two binomials to simplify the expression. The simplified expression is $x^2 + 17x + 60$, which is the same as the original expression. This demonstrates that factoring and simplifying quadratic expressions can be a powerful tool for solving equations and inequalities.

Common Mistakes to Avoid

When simplifying quadratic expressions, there are several common mistakes to avoid. These include:

• Not factoring the expression correctly
• Not multiplying the binomials correctly
• Not simplifying the expression correctly

To avoid these mistakes, it is essential to follow the steps outlined above and to double-check your work.

Real-World Applications

Simplifying quadratic expressions has 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. In economics, quadratic expressions are used to model the behavior of markets and economies.

Final Thoughts

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Introduction

In our previous article, we simplified the quadratic expression $x^2 + 17x + 60$ by factoring it into the form $(x + 5)(x + 12)$. We then multiplied the two binomials to simplify the expression. In this article, we will answer some common questions related to simplifying quadratic expressions.

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 more compact and manageable form. Factoring is a specific technique used to simplify quadratic expressions.

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

A: A quadratic expression can be factored if it can be expressed as a product of two binomials. To determine if a quadratic expression can be factored, we need to check if the expression can be written in the form $(x + a)(x + b)$, where $a$ and $b$ are constants.

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

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

• Not factoring the expression correctly
• Not multiplying the binomials correctly
• Not simplifying the expression correctly
• Not checking if the expression can be factored

Q: How do I simplify a quadratic expression that cannot be factored?

A: If a quadratic expression cannot be factored, we can simplify it using other techniques, such as completing the square or using the quadratic formula. Completing the square involves rewriting the quadratic expression in the form $(x + a)^2 + b$, while the quadratic formula involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the solutions to the quadratic equation.

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
• Modeling the behavior of markets and economies in economics

Q: How do I know if a quadratic expression is a perfect square trinomial?

A: A quadratic expression is a perfect square trinomial if it can be written in the form $(x + a)^2$, where $a$ is a constant. To determine if a quadratic expression is a perfect square trinomial, we need to check if the expression can be written in this form.

Q: What are some common types of quadratic expressions?

A: Some common types of quadratic expressions include:

• Monic quadratic expressions: quadratic expressions of the form $x^2 + bx + c$
• Non-monic quadratic expressions: quadratic expressions of the form $ax^2 + bx + c$, where $a \neq 1$
• Perfect square trinomials: quadratic expressions of the form $(x + a)^2$

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

A: To simplify a quadratic expression with a negative leading coefficient, we need to follow the same steps as for a quadratic expression with a positive leading coefficient. However, we need to be careful when multiplying the binomials, as the negative sign may affect the result.

Conclusion

In this article, we answered some common questions related to simplifying quadratic expressions. We discussed the difference between factoring and simplifying a quadratic expression, common mistakes to avoid, and real-world applications of simplifying quadratic expressions. We also discussed how to simplify quadratic expressions that cannot be factored and how to determine if a quadratic expression is a perfect square trinomial.