Use the method of finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term

Oct 13, 2025 by talynt.org 108 views

a) $a^2 + 9a + 8$

To factor the expression $a^2 + 9a + 8$ , we need to find two numbers whose product is $8$ and whose sum is $9$ . These numbers are $8$ and $1$ , because $8 \times 1 = 8$ and $8 + 1 = 9$ . Therefore, we can write the expression as $(a + 8)(a + 1)$ .

c) $c^2 + 10c + 24$

To factor the expression $c^2 + 10c + 24$ , we need to find two numbers whose product is $24$ and whose sum is $10$ . These numbers are $12$ and $2$ , because $12 \times 2 = 24$ and $12 + 2 = 14$ , which is not correct, but $6 \times 4 = 24$ and $6 + 4 = 10$ . Therefore, we can write the expression as $(c + 6)(c + 4)$ .

e) $x^2 - 18x + 72$

To factor the expression $x^2 - 18x + 72$ , we need to find two numbers whose product is $72$ and whose sum is $-18$ . These numbers are $-24$ and $-3$ , because $-24 \times -3 = 72$ and $-24 + (-3) = -27$ , which is not correct, but $-18 \times -4 = 72$ and $-18 + (-4) = -22$ , which is not correct, but $-12 \times -6 = 72$ and $-12 + (-6) = -18$ . Therefore, we can write the expression as $(x - 12)(x - 6)$ .

g) $z^2 - 13z + 36$

To factor the expression $z^2 - 13z + 36$ , we need to find two numbers whose product is $36$ and whose sum is $-13$ . These numbers are $-18$ and $-2$ , because $-18 \times -2 = 36$ and $-18 + (-2) = -20$ , which is not correct, but $-12 \times -3 = 36$ and $-12 + (-3) = -15$ , which is not correct, but $-9 \times -4 = 36$ and $-9 + (-4) = -13$ . Therefore, we can write the expression as $(z - 9)(z - 4)$ .

i) $l^2 + 7l - 30$

To factor the expression $l^2 + 7l - 30$ , we need to find two numbers whose product is $-30$ and whose sum is $7$ . These numbers are $-15$ and $2$ , because $-15 \times 2 = -30$ and $-15 + 2 = -13$ , which is not correct, but $-10 \times 3 = -30$ and $-10 + 3 = -7$ , which is not correct, but $-6 \times 5 = -30$ and $-6 + 5 = -1$ , which is not correct, but $-5 \times 6 = -30$ and $-5 + 6 = 1$ , which is not correct, but $-3 \times 10 = -30$ and $-3 + 10 = 7$ . Therefore, we can write the expression as $(l - 3)(l + 10)$ .

k) $n^2 - 4n - 5$

To factor the expression $n^2 - 4n - 5$ , we need to find two numbers whose product is $-5$ and whose sum is $-4$ . These numbers are $-5$ and $1$ , because $-5 \times 1 = -5$ and $-5 + 1 = -4$ . Therefore, we can write the expression as $(n - 5)(n + 1)$ .

Conclusion

In this section, we have factored several quadratic expressions using the method of finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term. This method is useful for factoring quadratic expressions that can be written in the form $(x + a)(x + b)$ or $(x - a)(x - b)$ . However, it is not always possible to factor a quadratic expression using this method, and in such cases, other methods such as the quadratic formula may be necessary.

Real-World Applications

Factoring quadratic expressions has many real-world applications, including:

Science and Engineering: Factoring quadratic expressions is used to solve problems in physics, engineering, and other sciences. For example, the motion of an object under the influence of gravity can be modeled using quadratic equations.
Economics: Factoring quadratic expressions is used to model economic systems and make predictions about future trends.
Computer Science: Factoring quadratic expressions is used in computer science to solve problems in algorithms and data structures.

Tips and Tricks

Here are some tips and tricks for factoring quadratic expressions:

Use the method of finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Use the quadratic formula to solve quadratic equations that cannot be factored.
Use a calculator or computer program to check your work and ensure that your answer is correct.

Practice Problems

Here are some practice problems to help you practice factoring quadratic expressions:

Factor the expression $x^2 + 5x + 6$ .
Factor the expression $y^2 - 7y + 12$ .
Factor the expression $z^2 + 2z - 15$ .

Solutions

Here are the solutions to the practice problems:

$x^2 + 5x + 6 = (x + 3)(x + 2)$ .
$y^2 - 7y + 12 = (y - 3)(y - 4)$ .
$z^2 + 2z - 15 = (z + 5)(z - 3)$ .

Conclusion

In this section, we have discussed how to factor quadratic expressions using the method of finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term. We have also discussed some real-world applications of factoring quadratic expressions and provided some tips and tricks for factoring quadratic expressions. Finally, we have provided some practice problems to help you practice factoring quadratic expressions.

Q: What is the first step in factoring a quadratic expression?

A: The first step in factoring a quadratic expression is to determine if the expression can be factored using the method of finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: How do I determine if a quadratic expression can be factored using the method of finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term?

A: To determine if a quadratic expression can be factored using the method of finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term.

Q: What if I am unable to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term?

A: If you are unable to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term, you may need to use other methods such as the quadratic formula to solve the quadratic equation.

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 given by:

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

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of $a$ , $b$ , and $c$ into the formula and simplify.

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

A: Factoring quadratic expressions has many real-world applications, including:

Science and Engineering: Factoring quadratic expressions is used to solve problems in physics, engineering, and other sciences. For example, the motion of an object under the influence of gravity can be modeled using quadratic equations.
Economics: Factoring quadratic expressions is used to model economic systems and make predictions about future trends.
Computer Science: Factoring quadratic expressions is used in computer science to solve problems in algorithms and data structures.

Q: What are some tips and tricks for factoring quadratic expressions?

A: Here are some tips and tricks for factoring quadratic expressions:

Use the method of finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Use the quadratic formula to solve quadratic equations that cannot be factored.
Use a calculator or computer program to check your work and ensure that your answer is correct.

Q: How can I practice factoring quadratic expressions?

A: You can practice factoring quadratic expressions by working through practice problems and checking your answers with a calculator or computer program.

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

A: Here are some common mistakes to avoid when factoring quadratic expressions:

Not checking your work: Make sure to check your work by plugging in the values of $x$ into the original equation and simplifying.
Not using the correct method: Make sure to use the correct method for factoring quadratic expressions, such as the method of finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term.
Not simplifying the expression: Make sure to simplify the expression by combining like terms.

Q: How can I use factoring to solve real-world problems?

A: You can use factoring to solve real-world problems by applying the concepts and techniques learned in this section to real-world scenarios. For example, you can use factoring to model the motion of an object under the influence of gravity, or to model economic systems and make predictions about future trends.

Q: What are some advanced topics in factoring quadratic expressions?

A: Some advanced topics in factoring quadratic expressions include:

Factoring quadratic expressions with complex coefficients: This involves factoring quadratic expressions with complex coefficients, such as $x^2 + 2ix + 1$ .
Factoring quadratic expressions with rational coefficients: This involves factoring quadratic expressions with rational coefficients, such as $x^2 + \frac{1}{2}x + \frac{1}{4}$ .
Factoring quadratic expressions with polynomial coefficients: This involves factoring quadratic expressions with polynomial coefficients, such as $x^2 + 2x^3 + x^4$ .

Q: How can I learn more about factoring quadratic expressions?

A: You can learn more about factoring quadratic expressions by:

Reading textbooks and online resources: There are many textbooks and online resources available that provide detailed explanations and examples of factoring quadratic expressions.
Watching video tutorials: There are many video tutorials available that provide step-by-step instructions and examples of factoring quadratic expressions.
Practicing with problems: The best way to learn about factoring quadratic expressions is to practice with problems and check your answers with a calculator or computer program.