Solve Using Factoring Or The Quadratic Formula: X 2 + 4 X − 32 = 0 X^2 + 4x - 32 = 0 X 2 + 4 X − 32 = 0 A. X = 9 X = 9 X = 9 And X = − 5 X = -5 X = − 5 B. X = − 10 X = -10 X = − 10 And X = − 4 X = -4 X = − 4 C. X = − 8 X = -8 X = − 8 And X = 4 X = 4 X = 4 D. X = − 4 X = -4 X = − 4 And X = 10 X = 10 X = 10

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore two methods for solving quadratic equations: factoring and the quadratic formula. We will apply these methods to a specific quadratic equation, $x^2 + 4x - 32 = 0$, and determine the correct solutions.

What are Quadratic Equations?

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. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

Factoring Quadratic Equations

Factoring is a method of solving quadratic equations by expressing them as a product of two binomials. The general form of a factored quadratic equation is:

$$(x + m)(x + n) = 0$$

where $m$ and $n$ are constants. To factor a quadratic equation, 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 ($b$).

Applying Factoring to the Given Equation

Let's apply factoring to the given equation, $x^2 + 4x - 32 = 0$. We need to find two numbers whose product is equal to $-32$ and whose sum is equal to $4$. After some trial and error, we find that the numbers are $8$ and $-4$, since $8 \times (-4) = -32$ and $8 + (-4) = 4$.

Therefore, we can write the factored form of the equation as:

$$(x + 8)(x - 4) = 0$$

Solving for $x$

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

$$x + 8 = 0 \Rightarrow x = -8$$

$$x - 4 = 0 \Rightarrow x = 4$$

Therefore, the solutions to the equation $x^2 + 4x - 32 = 0$ using factoring are $x = -8$ and $x = 4$.

The Quadratic Formula

The quadratic formula is a method of solving quadratic equations that is based on the fact that a quadratic equation can be written in the form:

$$ax^2 + bx + c = a(x - r_1)(x - r_2)$$

where $r_1$ and $r_2$ are the roots of the equation. The quadratic formula is:

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

Applying the Quadratic Formula to the Given Equation

Let's apply the quadratic formula to the given equation, $x^2 + 4x - 32 = 0$. We have:

$$a = 1, b = 4, c = -32$$

Plugging these values into the quadratic formula, we get:

$$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(-32)}}{2(1)}$$

Simplifying, we get:

$$x = \frac{-4 \pm \sqrt{16 + 128}}{2}$$

$$x = \frac{-4 \pm \sqrt{144}}{2}$$

$$x = \frac{-4 \pm 12}{2}$$

Therefore, the solutions to the equation $x^2 + 4x - 32 = 0$ using the quadratic formula are $x = -8$ and $x = 4$.

Conclusion

In this article, we have explored two methods for solving quadratic equations: factoring and the quadratic formula. We applied these methods to the equation $x^2 + 4x - 32 = 0$ and determined the correct solutions. The solutions to the equation are $x = -8$ and $x = 4$.

Answer

The correct answer is:

• C. $x = -8$ and $x = 4$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In our previous article, we explored two methods for solving quadratic equations: factoring and the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations and provide additional guidance on how to solve them.

Q&A

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.

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

A: A quadratic equation can be factored if it can be written in the form:

$$(x + m)(x + n) = 0$$

where $m$ and $n$ are constants. To factor a quadratic equation, you 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 ($b$).

Q: What is the quadratic formula?

A: The quadratic formula is a method of solving quadratic equations that is based on the fact that a quadratic equation can be written in the form:

$$ax^2 + bx + c = a(x - r_1)(x - r_2)$$

where $r_1$ and $r_2$ are the roots of the equation. The quadratic formula is:

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

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression and solve for $x$.

Q: What are the steps to solve a quadratic equation?

A: The steps to solve a quadratic equation are:

1. Write the equation in the form $ax^2 + bx + c = 0$.
2. Check if the equation can be factored.
3. If the equation can be factored, factor it and solve for $x$.
4. If the equation cannot be factored, use the quadratic formula to solve for $x$.
5. Simplify the expression and solve for $x$.

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

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

• Not checking if the equation can be factored before using the quadratic formula.
• Not simplifying the expression before solving for $x$.
• Not checking if the solutions are valid (i.e., not checking if the solutions are real numbers).

Additional Tips and Resources

• Practice, practice, practice! Solving quadratic equations takes practice, so make sure to practice regularly.
• Use online resources, such as Khan Academy or Mathway, to help you solve quadratic equations.
• Watch video tutorials or online lectures to help you understand the concepts better.
• Join a study group or find a study buddy to help you stay motivated and get help when you need it.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations and provided additional guidance on how to solve them. We hope that this article has been helpful in clarifying any confusion you may have had about quadratic equations. Remember to practice regularly and seek help when you need it. Good luck!