Writing the Equation in Factored Form

Oct 13, 2025 by miodeya.com 95 views

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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 the quadratic equation $16x^2 - 25 = 0$ by factoring. Factoring is a powerful method for solving quadratic equations, and it is essential to understand how to apply it correctly.

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

The Equation $16x^2 - 25 = 0$

The equation we will be solving is $16x^2 - 25 = 0$ . This equation is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 16, b = 0, and c = -25.

Factoring the Equation

To factor the equation $16x^2 - 25 = 0$ , we need to find two numbers whose product is -400 (16 * -25) and whose sum is 0 (since there is no middle term in the equation). These numbers are 20 and -20, since 20 * -20 = -400 and 20 + (-20) = 0.

Writing the Equation in Factored Form

Now that we have found the numbers 20 and -20, we can write the equation in factored form:

(4x - 5)(4x + 5) = 0

Solving for x

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

4x - 5 = 0 --> 4x = 5 --> x = 5/4

4x + 5 = 0 --> 4x = -5 --> x = -5/4

Conclusion

In this article, we have solved the quadratic equation $16x^2 - 25 = 0$ by factoring. We have shown that the equation can be written in factored form as (4x - 5)(4x + 5) = 0, and we have solved for x by setting each factor equal to zero. Factoring is a powerful method for solving quadratic equations, and it is essential to understand how to apply it correctly.

Tips and Tricks

When factoring a quadratic equation, look for two numbers whose product is the constant term (in this case, -400) and whose sum is the coefficient of the middle term (in this case, 0).
Use the distributive property to expand the factored form of the equation and verify that it is equivalent to the original equation.
Check your solutions by plugging them back into the original equation to make sure they are true.

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, friction, and other forces.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Quadratic equations are used to model economic systems, including supply and demand curves and investment portfolios.

Common Mistakes

Not checking solutions: Make sure to check your solutions by plugging them back into the original equation to make sure they are true.
Not using the distributive property: Use the distributive property to expand the factored form of the equation and verify that it is equivalent to the original equation.
Not looking for two numbers: When factoring a quadratic equation, look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term.

Conclusion

In conclusion, solving quadratic equations by factoring is a powerful method that can be used to solve a wide range of equations. By following the steps outlined in this article, you can learn how to factor quadratic equations and solve for x. Remember to check your solutions, use the distributive property, and look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term. With practice and patience, you can become proficient in solving quadratic equations by factoring.

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Introduction

In our previous article, we discussed how to solve quadratic equations by factoring. However, we know that practice makes perfect, and there's no better way to learn than by asking questions and getting answers. In this article, we'll provide a Q&A section to help you better understand how to factor quadratic equations and solve for x.

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

A: The first step in factoring a quadratic equation is to look for two numbers whose product is the constant term (in this case, -400) and whose sum is the coefficient of the middle term (in this case, 0).

Q: How do I find the two numbers?

A: To find the two numbers, you can use the following steps:

Multiply the constant term (in this case, -400) by the coefficient of the middle term (in this case, 0).
Look for two numbers whose product is the result of step 1 and whose sum is the coefficient of the middle term (in this case, 0).

Q: What if I don't find two numbers?

A: If you don't find two numbers, it may be because the equation is not factorable. In this case, you can use other methods, such as the quadratic formula, to solve the equation.

Q: How do I write the equation in factored form?

A: To write the equation in factored form, you can use the two numbers you found in the previous step. For example, if the two numbers are 20 and -20, you can write the equation as (4x - 5)(4x + 5) = 0.

Q: How do I solve for x?

A: To solve for x, you need to set each factor equal to zero and solve for x. For example, if the equation is (4x - 5)(4x + 5) = 0, you can set each factor equal to zero and solve for x:

4x - 5 = 0 --> 4x = 5 --> x = 5/4

4x + 5 = 0 --> 4x = -5 --> x = -5/4

Q: What if I get multiple solutions?

A: If you get multiple solutions, it means that the equation has multiple roots. In this case, you need to check each solution to make sure it is true.

Q: Can I use factoring to solve quadratic equations with complex numbers?

A: Yes, you can use factoring to solve quadratic equations with complex numbers. However, you need to be careful when working with complex numbers, as they can be tricky to handle.

Q: Are there any tips and tricks for factoring quadratic equations?

A: Yes, here are some tips and tricks for factoring quadratic equations:

Use the distributive property to expand the factored form of the equation and verify that it is equivalent to the original equation.
Check your solutions by plugging them back into the original equation to make sure they are true.
Look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term.

Q: Can I use factoring to solve quadratic equations with negative coefficients?

A: Yes, you can use factoring to solve quadratic equations with negative coefficients. However, you need to be careful when working with negative coefficients, as they can change the sign of the equation.

Q: Are there any real-world applications of factoring quadratic equations?

A: Yes, there are many real-world applications of factoring quadratic equations, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
Economics: Quadratic equations are used to model economic systems, including supply and demand curves and investment portfolios.

Conclusion

In conclusion, factoring quadratic equations is a powerful method that can be used to solve a wide range of equations. By following the steps outlined in this article, you can learn how to factor quadratic equations and solve for x. Remember to check your solutions, use the distributive property, and look for two numbers whose product is the constant term and whose sum is the coefficient of the middle term. With practice and patience, you can become proficient in solving quadratic equations by factoring.