Factor The Quadratic Expression: ${ 5x^2 + 34x + 24 }$

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Introduction

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Quadratic expressions are a fundamental concept in mathematics, and factoring them is a crucial skill to master. In this article, we will delve into the world of quadratic expressions and explore the process of factoring them. We will focus on the quadratic expression $5x^2 + 34x + 24$ and provide a step-by-step guide on how to factor it.

What is a Quadratic Expression?

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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 factored into the product of two binomials, which is a fundamental concept in algebra.

Why Factor Quadratic Expressions?

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Factoring quadratic expressions is an essential skill in mathematics, and it has numerous applications in various fields, such as physics, engineering, and economics. By factoring quadratic expressions, we can:

• Simplify complex expressions
• Solve quadratic equations
• Find the roots of a quadratic equation
• Understand the behavior of quadratic functions

The Factorization Process

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The process of factoring a quadratic expression involves finding two binomials whose product is equal to the original expression. To factor a quadratic expression, we need to follow these steps:

1. Check if the expression can be factored by grouping: If the expression can be grouped into two parts, we can factor it by grouping.
2. Check if the expression can be factored by finding the greatest common factor (GCF): If the expression has a common factor, we can factor it out.
3. Use the quadratic formula to find the roots: If the expression cannot be factored, we can use the quadratic formula to find the roots.

Factoring the Quadratic Expression: $5x^2 + 34x + 24$

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Now, let's apply the factorization process to the quadratic expression $5x^2 + 34x + 24$. We will follow the steps outlined above to factor this expression.

Step 1: Check if the expression can be factored by grouping

To check if the expression can be factored by grouping, we need to look for two parts that can be grouped together. In this case, we can group the first two terms together: $5x^2 + 34x$. We can then look for a common factor in this group.

import sympy as sp
x = sp.symbols('x')

expr = 5x**2 + 34x + 24

factored_expr = sp.factor(expr)
print(factored_expr)

Step 2: Check if the expression can be factored by finding the GCF

To check if the expression can be factored by finding the GCF, we need to look for a common factor in the entire expression. In this case, we can see that the expression has a common factor of 1.

Step 3: Use the quadratic formula to find the roots

Since the expression cannot be factored by grouping or finding the GCF, we can use the quadratic formula to find the roots. The quadratic formula is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this case, $a = 5$, $b = 34$, and $c = 24$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-34 \pm \sqrt{34^2 - 4(5)(24)}}{2(5)}$$

Simplifying this expression, we get:

$$x = \frac{-34 \pm \sqrt{1156 - 480}}{10}$$

$$x = \frac{-34 \pm \sqrt{676}}{10}$$

$$x = \frac{-34 \pm 26}{10}$$

Therefore, the roots of the quadratic equation are:

$$x = \frac{-34 + 26}{10} = -\frac{8}{10} = -\frac{4}{5}$$

$$x = \frac{-34 - 26}{10} = -\frac{60}{10} = -6$$

Conclusion

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In this article, we have explored the process of factoring quadratic expressions. We have applied the factorization process to the quadratic expression $5x^2 + 34x + 24$ and found its roots using the quadratic formula. Factoring quadratic expressions is an essential skill in mathematics, and it has numerous applications in various fields. By mastering this skill, we can simplify complex expressions, solve quadratic equations, and understand the behavior of quadratic functions.

Final Answer

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The final answer is: $\boxed{(5x+6)(x+4)}$

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Introduction

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In our previous article, we explored the process of factoring quadratic expressions. We applied the factorization process to the quadratic expression $5x^2 + 34x + 24$ and found its roots using the quadratic formula. In this article, we will provide a Q&A guide to help you better understand the concept of factoring quadratic expressions.

Q: What is a quadratic expression?

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A: 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.

Q: Why factor quadratic expressions?

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A: Factoring quadratic expressions is an essential skill in mathematics, and it has numerous applications in various fields, such as physics, engineering, and economics. By factoring quadratic expressions, we can:

• Simplify complex expressions
• Solve quadratic equations
• Find the roots of a quadratic equation
• Understand the behavior of quadratic functions

Q: How do I factor a quadratic expression?

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A: To factor a quadratic expression, you need to follow these steps:

1. Check if the expression can be factored by grouping: If the expression can be grouped into two parts, you can factor it by grouping.
2. Check if the expression can be factored by finding the greatest common factor (GCF): If the expression has a common factor, you can factor it out.
3. Use the quadratic formula to find the roots: If the expression cannot be factored, you can use the quadratic formula to find the roots.

Q: What is the quadratic formula?

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A: The quadratic formula is given by:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In this formula, $a$, $b$, and $c$ are the coefficients of the quadratic expression, and $x$ is the variable.

Q: How do I use the quadratic formula to find the roots of a quadratic equation?

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A: To use the quadratic formula to find the roots of a quadratic equation, you need to follow these steps:

1. Plug in the values of $a$, $b$, and $c$: Plug in the values of $a$, $b$, and $c$ into the quadratic formula.
2. Simplify the expression: Simplify the expression under the square root.
3. Find the roots: Find the roots of the quadratic equation by solving for $x$.

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

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A: When factoring quadratic expressions, there are several common mistakes to avoid:

• Not checking if the expression can be factored by grouping: Make sure to check if the expression can be factored by grouping before trying to factor it by finding the GCF.
• Not finding the greatest common factor (GCF): Make sure to find the GCF of the expression before trying to factor it.
• Not using the quadratic formula correctly: Make sure to use the quadratic formula correctly to find the roots of the quadratic equation.

Q: How do I check if a quadratic expression can be factored by grouping?

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A: To check if a quadratic expression can be factored by grouping, you need to look for two parts that can be grouped together. In this case, you can group the first two terms together and look for a common factor.

Q: How do I find the greatest common factor (GCF) of a quadratic expression?

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A: To find the GCF of a quadratic expression, you need to look for a common factor in the entire expression. In this case, you can see that the expression has a common factor of 1.

Conclusion

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In this article, we have provided a Q&A guide to help you better understand the concept of factoring quadratic expressions. We have covered topics such as the definition of a quadratic expression, why factoring quadratic expressions is important, and how to factor a quadratic expression. We have also covered common mistakes to avoid when factoring quadratic expressions and how to check if a quadratic expression can be factored by grouping.

Final Answer

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The final answer is: $\boxed{(5x+6)(x+4)}$