Use Factoring To Solve The Following Quadratic Equation:$\[ 5x^2 = 27 - 42x \\]The Solution Set Is \{ \_\_ \}. (Type An Integer Or A Simplified Fraction. Use A Comma To Separate Multiple Solutions.)

Introduction

Quadratic equations are a fundamental concept in mathematics, and factoring is a powerful technique used to solve them. In this article, we will explore how to use factoring to solve a quadratic equation, and we will apply this technique to the given equation: $5x^2 = 27 - 42x$. We will also discuss the solution set and provide a step-by-step guide on how to factor the equation.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means 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.

The Given Equation

The given equation is:

$$5x^2 = 27 - 42x$$

To solve this equation using factoring, we need to rewrite it in the standard form of a quadratic equation:

$$5x^2 + 42x - 27 = 0$$

Step 1: Factor the Equation

To factor the equation, we need to find two numbers whose product is $5 \times (-27) = -135$ and whose sum is $42$. These numbers are $45$ and $-3$, since $45 \times (-3) = -135$ and $45 + (-3) = 42$.

So, we can rewrite the equation as:

$$5x^2 + 45x - 3x - 27 = 0$$

Now, we can factor the equation by grouping:

$$(5x^2 + 45x) + (-3x - 27) = 0$$

$$(5x(x + 9)) + (-3(x + 9)) = 0$$

$$(5x - 3)(x + 9) = 0$$

Step 2: Solve for x

To solve for x, we need to set each factor equal to zero and solve for x:

$$(5x - 3) = 0 \Rightarrow 5x = 3 \Rightarrow x = \frac{3}{5}$$

$$(x + 9) = 0 \Rightarrow x = -9$$

The Solution Set

The solution set is the set of all possible values of x that satisfy the equation. In this case, the solution set is:

$$\left\{\frac{3}{5}, -9\right\}$$

Conclusion

In this article, we used factoring to solve a quadratic equation. We first rewrote the equation in the standard form, then factored the equation by grouping, and finally solved for x by setting each factor equal to zero. The solution set is the set of all possible values of x that satisfy the equation. We hope this article has provided a clear and concise explanation of how to use factoring to solve quadratic equations.

Tips and Tricks

• When factoring a quadratic equation, look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
• Use the distributive property to factor the equation by grouping.
• Set each factor equal to zero and solve for x to find the solution set.

Common Mistakes

• Not rewriting the equation in the standard form.
• Not factoring the equation correctly.
• Not solving for x correctly.

Real-World Applications

Quadratic equations have many real-world applications, such as:

• Modeling the trajectory of a projectile.
• Finding the maximum or minimum value of a function.
• Solving problems in physics, engineering, and economics.

Further Reading

For further reading on quadratic equations and factoring, we recommend the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Factoring Quadratic Equations
• Wolfram Alpha: Quadratic Equation Solver
  Quadratic Equations: A Q&A Guide =====================================

Introduction

Quadratic equations are a fundamental concept in mathematics, and factoring is a powerful technique used to solve them. In this article, we will provide a Q&A guide on quadratic equations and factoring, covering common questions and topics.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means 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.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and graphing. Factoring is a powerful technique used to solve quadratic equations by expressing the equation as a product of two binomials.

Q: What is factoring?

A: Factoring is a technique used to express a quadratic equation as a product of two binomials. It involves finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, follow these steps:

1. Rewrite the equation in the standard form.
2. Look for two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
3. Use the distributive property to factor the equation by grouping.
4. Set each factor equal to zero and solve for x.

Q: What is the quadratic formula?

A: The quadratic formula is a method used to solve quadratic equations by expressing the equation in the form:

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

where a, b, and c are constants.

Q: When should I use the quadratic formula?

A: Use the quadratic formula when the equation cannot be factored easily or when the equation is in the form of a quadratic expression.

Q: What is the difference between factoring and the quadratic formula?

A: Factoring is a technique used to express a quadratic equation as a product of two binomials, while the quadratic formula is a method used to solve quadratic equations by expressing the equation in a specific form.

Q: Can I use both factoring and the quadratic formula to solve a quadratic equation?

A: Yes, you can use both factoring and the quadratic formula to solve a quadratic equation. However, factoring is often a more efficient and easier method to use.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not rewriting the equation in the standard form.
• Not factoring the equation correctly.
• Not solving for x correctly.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, follow these steps:

1. Check if the equation can be factored easily.
2. If the equation can be factored, check if the factors have a common factor.
3. If the factors have a common factor, the equation has one solution.
4. If the factors do not have a common factor, the equation has two solutions.

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

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

• Modeling the trajectory of a projectile.
• Finding the maximum or minimum value of a function.
• Solving problems in physics, engineering, and economics.

Conclusion

In this article, we provided a Q&A guide on quadratic equations and factoring, covering common questions and topics. We hope this guide has provided a clear and concise explanation of how to solve quadratic equations using factoring and the quadratic formula.