Which Quadratic Equation Is Equivalent To ( X 2 − 1 ) 2 − 11 ( X 2 − 1 ) + 24 = 0 \left(x^2-1\right)^2-11\left(x^2-1\right)+24=0 ( X 2 − 1 ) 2 − 11 ( X 2 − 1 ) + 24 = 0 ?A. U 2 − 11 U + 24 = 0 U^2-11u+24=0 U 2 − 11 U + 24 = 0 Where U=\left(x^2-1\right ]B. ( U 2 ) 2 − 11 ( U 2 ) + 24 \left(u^2\right)^2-11\left(u^2\right)+24 ( U 2 ) 2 − 11 ( U 2 ) + 24 Where

Mar 8, 2025 by ADMIN 359 views

**Solving Quadratic Equations: A Step-by-Step Guide**

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will explore the concept of quadratic equations and provide a step-by-step guide on how to solve them. We will also discuss the different types of quadratic equations and how to identify the equivalent forms of a given quadratic equation.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (usually 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.

Types of Quadratic Equations

There are several types of quadratic equations, including:

Monic quadratic equations: These are quadratic equations where the coefficient of the x^2 term is 1. For example: x^2 + 3x + 2 = 0
Non-monic quadratic equations: These are quadratic equations where the coefficient of the x^2 term is not 1. For example: 2x^2 + 3x + 1 = 0
Perfect square quadratic equations: These are quadratic equations that can be factored as a perfect square. For example: x^2 + 4x + 4 = (x + 2)^2 = 0

Equivalent Forms of Quadratic Equations

A quadratic equation can have multiple equivalent forms, which can be obtained by performing algebraic manipulations. For example:

Factoring: A quadratic equation can be factored into the product of two binomials. For example: x^2 + 5x + 6 = (x + 3)(x + 2) = 0
Completing the square: A quadratic equation can be rewritten in the form (x + a)^2 + b = 0, where a and b are constants. For example: x^2 + 4x + 4 = (x + 2)^2 = 0
Substitution: A quadratic equation can be rewritten in terms of a new variable, which can simplify the equation. For example: x^2 - 4x + 4 = (x - 2)^2 = 0

Solving the Given Quadratic Equation

The given quadratic equation is:

$\left(x^2-1\right)^2-11\left(x^2-1\right)+24=0$

To solve this equation, we can start by expanding the left-hand side:

$\left(x^2-1\right)^2-11\left(x^2-1\right)+24=0$

$\Rightarrow x^4-2x^2+1-11x^2+11+24=0$

$\Rightarrow x^4-13x^2+36=0$

Factoring the Quadratic Equation

We can factor the quadratic equation as:

$x^4-13x^2+36=0$

$\Rightarrow (x^2-9)(x^2-4)=0$

Solving for x

We can solve for x by setting each factor equal to zero:

$x^2-9=0$

$\Rightarrow x^2=9$

$\Rightarrow x=\pm 3$

$x^2-4=0$

$\Rightarrow x^2=4$

$\Rightarrow x=\pm 2$

Conclusion

In this article, we have discussed the concept of quadratic equations and provided a step-by-step guide on how to solve them. We have also discussed the different types of quadratic equations and how to identify the equivalent forms of a given quadratic equation. Finally, we have solved the given quadratic equation and obtained the solutions for x.

Which Quadratic Equation is Equivalent to the Given Quadratic Equation?

The given quadratic equation is:

$\left(x^2-1\right)^2-11\left(x^2-1\right)+24=0$

We have factored the quadratic equation as:

$x^4-13x^2+36=0$

$\Rightarrow (x^2-9)(x^2-4)=0$

We can see that the quadratic equation can be rewritten in the form:

$u^2-11u+24=0$

where $u=\left(x^2-1\right)$

Therefore, the correct answer is:

A. $u^2-11u+24=0$ where $u=\left(x^2-1\right)$

Discussion

The given quadratic equation can be rewritten in the form:

$u^2-11u+24=0$

where $u=\left(x^2-1\right)$

This is an example of a substitution method, where we have replaced the variable x with a new variable u.

The quadratic equation can also be rewritten in the form:

$\left(u^2\right)^2-11\left(u^2\right)+24$

where $u^2=\left(x^2-1\right)$

This is an example of a substitution method, where we have replaced the variable x with a new variable u.

However, the correct answer is:

A. $u^2-11u+24=0$ where $u=\left(x^2-1\right)$

This is because the quadratic equation can be factored as:

$x^4-13x^2+36=0$

$\Rightarrow (x^2-9)(x^2-4)=0$

We can see that the quadratic equation can be rewritten in the form:

$u^2-11u+24=0$

where $u=\left(x^2-1\right)$

Therefore, the correct answer is:

A. $u^2-11u+24=0$ where $u=\left(x^2-1\right)$

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Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will provide a Q&A guide on quadratic equations, covering topics such as solving quadratic equations, factoring, and substitution.

Q: What is a Quadratic Equation?

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable (usually 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: To solve a quadratic equation, you can use various methods, including factoring, completing the square, and substitution. Here are the steps to solve a quadratic equation:

Check if the equation can be factored: If the equation can be factored, you can use the factoring method to solve it.
Use the quadratic formula: If the equation cannot be factored, you can use the quadratic formula to solve it. The quadratic formula is:

x = (-b ± √(b^2 - 4ac)) / 2a

Complete the square: If the equation is in the form ax^2 + bx + c = 0, you can complete the square to solve it.

Q: What is Factoring?

A: Factoring is a method of solving quadratic equations by expressing the equation as a product of two binomials. For example:

x^2 + 5x + 6 = (x + 3)(x + 2) = 0

Q: What is Completing the Square?

A: Completing the square is a method of solving quadratic equations by rewriting the equation in the form (x + a)^2 + b = 0. For example:

x^2 + 4x + 4 = (x + 2)^2 = 0

Q: What is Substitution?

A: Substitution is a method of solving quadratic equations by replacing the variable x with a new variable u. For example:

x^2 - 4x + 4 = (x - 2)^2 = 0

Q: How Do I Determine if a Quadratic Equation Can be Factored?

A: To determine if a quadratic equation can be factored, you can try to factor the equation by finding two binomials whose product is equal to the original equation. You can also use the factoring method to check if the equation can be factored.

Q: What is the Quadratic Formula?

A: The quadratic formula is a method of solving quadratic equations by using the formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Q: How Do I Use the Quadratic Formula?

A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. You can then simplify the expression to find the solutions to the equation.

Q: What is the Difference Between a Monic Quadratic Equation and a Non-Monic Quadratic Equation?

A: A monic quadratic equation is a quadratic equation where the coefficient of the x^2 term is 1. For example:

x^2 + 3x + 2 = 0

A non-monic quadratic equation is a quadratic equation where the coefficient of the x^2 term is not 1. For example:

2x^2 + 3x + 1 = 0

Q: How Do I Solve a Perfect Square Quadratic Equation?

A: To solve a perfect square quadratic equation, you can use the factoring method to rewrite the equation as a perfect square. For example:

x^2 + 4x + 4 = (x + 2)^2 = 0

Q: What is the Importance of Quadratic Equations?

A: Quadratic equations are important in various fields, including physics, engineering, and economics. They are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of financial markets.

Conclusion

In this article, we have provided a Q&A guide on quadratic equations, covering topics such as solving quadratic equations, factoring, and substitution. We hope that this guide has been helpful in understanding the concept of quadratic equations and how to solve them.