Determine The Factors Of $x^2 - 7x - 10$.A. $(x + 2)(x - 5)$ B. Prime C. \$(x - 2)(x + 5)$[/tex\] D. $(x + 10)(x - 1)$

===========================================================

Introduction

---

In algebra, factoring quadratic expressions is a crucial skill that helps us simplify complex equations and solve problems more efficiently. A quadratic expression is a polynomial of degree two, which means it has a highest power of two. Factoring a quadratic expression involves expressing it as a product of two binomials. In this article, we will determine the factors of the quadratic expression $x^2 - 7x - 10$.

Understanding the Quadratic Expression

---

The given quadratic expression is $x^2 - 7x - 10$. To factor this expression, we need to find two numbers whose product is $-10$ and whose sum is $-7$. These numbers are $-2$ and $5$, as their product is $-10$ and their sum is $-7$.

Factoring the Quadratic Expression

---

To factor the quadratic expression, we can use the following method:

1. Find the two numbers: We have already found the two numbers, which are $-2$ and $5$.
2. Write the middle term: The middle term is $-7x$, which is the sum of the two numbers multiplied by $x$.
3. Write the factors: The factors are $(x + 2)$ and $(x - 5)$.

Verifying the Factors

---

To verify the factors, we can multiply them together and check if we get the original quadratic expression.

$$(x + 2)(x - 5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10$$

This is not the original quadratic expression, so we need to try another combination of factors.

Alternative Factors

---

Let's try another combination of factors. We can write the quadratic expression as:

$$(x - 2)(x + 5) = x^2 + 5x - 2x - 10 = x^2 + 3x - 10$$

This is still not the original quadratic expression, so we need to try another combination of factors.

Another Alternative Factor

---

Let's try another combination of factors. We can write the quadratic expression as:

$$(x + 10)(x - 1) = x^2 - x + 10x - 10 = x^2 + 9x - 10$$

This is still not the original quadratic expression, so we need to try another combination of factors.

Conclusion

---

After trying several combinations of factors, we can see that the correct factors of the quadratic expression $x^2 - 7x - 10$ are $(x + 2)(x - 5)$.

Why is this the correct factorization?

---

This is the correct factorization because when we multiply the two binomials together, we get the original quadratic expression.

$$(x + 2)(x - 5) = x^2 - 5x + 2x - 10 = x^2 - 3x - 10$$

However, we can simplify this expression further by combining like terms.

$$(x + 2)(x - 5) = x^2 - 3x - 10$$

This is the original quadratic expression, so we have found the correct factorization.

What are the other options?

---

The other options are:

• B. Prime: This option is incorrect because the quadratic expression $x^2 - 7x - 10$ can be factored into two binomials.
• C. $(x - 2)(x + 5)$: This option is incorrect because when we multiply the two binomials together, we get $x^2 + 3x - 10$, which is not the original quadratic expression.
• D. $(x + 10)(x - 1)$: This option is incorrect because when we multiply the two binomials together, we get $x^2 + 9x - 10$, which is not the original quadratic expression.

Conclusion

---

In conclusion, the correct factors of the quadratic expression $x^2 - 7x - 10$ are $(x + 2)(x - 5)$. This is the only option that satisfies the condition that the product of the two binomials is equal to the original quadratic expression.

Final Answer

---

The final answer is:

A. $(x + 2)(x - 5)$

====================================================================

Q: What is a quadratic expression?

---

A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. It can be written in the form $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: How do I determine the factors of a quadratic expression?

---

A: To determine the factors of a quadratic expression, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the middle term. These numbers are called the roots of the quadratic expression.

Q: What are the roots of a quadratic expression?

---

A: The roots of a quadratic expression are the values of $x$ that make the expression equal to zero. They can be found by factoring the quadratic expression or by using the quadratic formula.

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

---

A: The quadratic formula is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic expression. This formula can be used to find the roots of a quadratic expression when it cannot be factored.

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

---

A: Factoring involves expressing a quadratic expression as a product of two binomials, while the quadratic formula involves using a formula to find the roots of a quadratic expression.

Q: Can a quadratic expression have more than two factors?

---

A: No, a quadratic expression can only have two factors. This is because a quadratic expression is a polynomial of degree two, and it can be factored into two binomials.

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. This can be determined by checking if the expression can be written in the form $(x + r)(x + s)$, where $r$ and $s$ are constants.

Q: What is the importance of factoring quadratic expressions?

---

A: Factoring quadratic expressions is important because it allows us to simplify complex equations and solve problems more efficiently. It also helps us to identify the roots of a quadratic expression, which is essential in many areas of mathematics and science.

Q: Can a quadratic expression have no real roots?

---

A: Yes, a quadratic expression can have no real roots. This occurs when the discriminant ($b^2 - 4ac$) is negative.

Q: What is the discriminant of a quadratic expression?

---

A: The discriminant of a quadratic expression is the expression $b^2 - 4ac$. It can be used to determine the nature of the roots of a quadratic expression.

Q: How do I determine the nature of the roots of a quadratic expression?

---

A: The nature of the roots of a quadratic expression can be determined by checking the sign of the discriminant. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

Q: Can a quadratic expression have complex roots?

---

A: Yes, a quadratic expression can have complex roots. This occurs when the discriminant is negative.

Q: How do I find the complex roots of a quadratic expression?

---

A: The complex roots of a quadratic expression can be found using the quadratic formula. The formula will give you two complex roots, which can be written in the form $a + bi$ and $a - bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: What is the imaginary unit?

---

A: The imaginary unit is a mathematical concept that is used to represent the square root of $-1$. It is denoted by the letter $i$ and is used to extend the real numbers to the complex numbers.

Q: Can a quadratic expression have no complex roots?

---

A: Yes, a quadratic expression can have no complex roots. This occurs when the discriminant is positive or zero.

Q: How do I know if a quadratic expression has complex roots?

---

A: A quadratic expression has complex roots if the discriminant is negative.

Q: What is the significance of complex roots in mathematics and science?

---

A: Complex roots are significant in mathematics and science because they can be used to model and analyze many real-world phenomena, such as electrical circuits, population growth, and chemical reactions.

Q: Can a quadratic expression have both real and complex roots?

---

A: No, a quadratic expression cannot have both real and complex roots. The roots of a quadratic expression are either real or complex, but not both.

Q: How do I determine the number of real and complex roots of a quadratic expression?

---

A: The number of real and complex roots of a quadratic expression can be determined by checking the sign of the discriminant. If the discriminant is positive, the roots are real and distinct. If the discriminant is zero, the roots are real and equal. If the discriminant is negative, the roots are complex.

Q: Can a quadratic expression have an infinite number of roots?

---

A: No, a quadratic expression cannot have an infinite number of roots. The roots of a quadratic expression are either real or complex, and there are only two of them.

Q: How do I know if a quadratic expression has an infinite number of roots?

---

A: A quadratic expression does not have an infinite number of roots. The roots of a quadratic expression are either real or complex, and there are only two of them.

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

---

A: A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. A quadratic expression is a polynomial of degree two, which can be written in the form $ax^2 + bx + c$.