Consider The Following Quadratic Equation: 7 X 2 = 15 X − 2 7x^2 = 15x - 2 7 X 2 = 15 X − 2 Step 2 Of 2: Solve The Quadratic Equation By Factoring. Write Your Answer In Reduced Fraction Form, If Necessary.Answer: X = □ , X = □ X = \square, X = \square X = □ , X = □

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving quadratic equations by factoring, a method that involves expressing the quadratic equation as a product of two binomials. We will use the quadratic equation $7x^2 = 15x - 2$ as a case study to demonstrate the step-by-step process of solving quadratic equations by factoring.

Understanding Quadratic Equations

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

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $x$ is the variable. In our case, the quadratic equation is $7x^2 = 15x - 2$, which can be rewritten as:

$$7x^2 - 15x + 2 = 0$$

Step 1: Rearrange the Equation

To solve the quadratic equation by factoring, we need to rearrange the equation to get it in the standard form:

$$ax^2 + bx + c = 0$$

In our case, we need to move the constant term to the left-hand side of the equation:

$$7x^2 - 15x + 2 = 0$$

Step 2: Factor the Quadratic Equation

Now that we have the equation in the standard form, we can try to factor it. Factoring a quadratic equation involves expressing it as a product of two binomials. To do this, we need to find two numbers whose product is $ac$ (in this case, $7 \times 2 = 14$) and whose sum is $b$ (in this case, $-15$).

After some trial and error, we find that the two numbers are $-14$ and $-1$. Therefore, we can write the quadratic equation as:

$$(7x^2 - 14x) + (x - 2) = 0$$

Now, we can factor out the greatest common factor (GCF) from each term:

$$7x(x - 2) + 1(x - 2) = 0$$

Step 3: Solve for x

Now that we have factored the quadratic equation, we can solve for $x$. To do this, we need to set each factor equal to zero and solve for $x$.

Setting the first factor equal to zero, we get:

$$7x(x - 2) = 0$$

This gives us two possible solutions:

$$x = 0 \quad \text{or} \quad x - 2 = 0$$

Solving the second equation, we get:

$$x = 2$$

Therefore, the solutions to the quadratic equation are:

$$x = 0 \quad \text{or} \quad x = 2$$

Conclusion

In this article, we have demonstrated the step-by-step process of solving quadratic equations by factoring. We used the quadratic equation $7x^2 = 15x - 2$ as a case study and showed how to factor it and solve for $x$. By following these steps, you can solve quadratic equations by factoring and gain a deeper understanding of this important mathematical concept.

Common Mistakes to Avoid

When solving quadratic equations by factoring, there are several common mistakes to avoid. These include:

• Not rearranging the equation: Make sure to rearrange the equation to get it in the standard form.
• Not factoring correctly: Make sure to factor the quadratic equation correctly, using the correct numbers and signs.
• Not solving for x: Make sure to set each factor equal to zero and solve for $x$.

By avoiding these common mistakes, you can ensure that you are solving quadratic equations by factoring correctly and accurately.

Real-World Applications

Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

By understanding how to solve quadratic equations by factoring, you can gain a deeper understanding of these real-world applications and make more informed decisions.

Final Thoughts

Introduction

In our previous article, we demonstrated the step-by-step process of solving quadratic equations by factoring. In this article, we will answer some of the most frequently asked questions about quadratic equations by factoring.

Q: What is a quadratic equation?

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

$$ax^2 + bx + c = 0$$

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

A: To determine if a quadratic equation can be factored, you need to check if the equation can be written as a product of two binomials. If the equation can be written in this form, then it can be factored.

Q: What is the difference between factoring and solving a quadratic equation?

A: Factoring a quadratic equation involves expressing it as a product of two binomials, while solving a quadratic equation involves finding the values of the variable (in this case, $x$) that satisfy the equation.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two numbers whose product is $ac$ (in this case, $a \times c$) and whose sum is $b$ (in this case, $b$). Once you have found these numbers, you can write the quadratic equation as a product of two binomials.

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

A: Some common mistakes to avoid when factoring a quadratic equation include:

• Not rearranging the equation: Make sure to rearrange the equation to get it in the standard form.
• Not factoring correctly: Make sure to factor the quadratic equation correctly, using the correct numbers and signs.
• Not solving for x: Make sure to set each factor equal to zero and solve for $x$.

Q: Can all quadratic equations be factored?

A: No, not all quadratic equations can be factored. Some quadratic equations may not be able to be factored, and in these cases, other methods such as the quadratic formula may need to be used.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations that cannot be factored. The quadratic formula is:

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

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when a quadratic equation cannot be factored. The quadratic formula is a powerful tool that can be used to solve quadratic equations that cannot be factored.

Q: Can I use the quadratic formula to solve quadratic equations that can be factored?

A: Yes, you can use the quadratic formula to solve quadratic equations that can be factored. However, in these cases, it is usually more efficient to factor the quadratic equation and solve for $x$.

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

A: Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic equations by factoring. We have covered topics such as the definition of a quadratic equation, how to determine if a quadratic equation can be factored, and how to factor a quadratic equation. We have also discussed some common mistakes to avoid when factoring a quadratic equation and when to use the quadratic formula. By understanding these concepts, you can gain a deeper understanding of quadratic equations and develop the skills you need to solve them.

Additional Resources

For more information on quadratic equations and factoring, we recommend the following resources:

• Textbooks: "Algebra and Trigonometry" by Michael Sullivan and "College Algebra" by James Stewart
• Online resources: Khan Academy, Mathway, and Wolfram Alpha
• Practice problems: Quadratic equation practice problems on Mathway and Wolfram Alpha

By using these resources, you can gain a deeper understanding of quadratic equations and develop the skills you need to solve them.