Identifying the Quadratic Equation

Oct 13, 2025 by windailys.com 52 views

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. In this article, we will focus on identifying the quadratic equation that has solutions of $\frac{-2}{3}$ and 4.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which can be written in the general form:

ax^2 + bx + c = 0

where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. The solutions to a quadratic equation are the values of $x$ that satisfy the equation.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$ are given by:

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

Identifying the Quadratic Equation

To identify the quadratic equation that has solutions of $\frac{-2}{3}$ and 4, we need to use the fact that if $r$ is a solution to a quadratic equation, then $(x - r)$ is a factor of the equation. Therefore, we can write the quadratic equation as:

y = a(x - \frac{-2}{3})(x - 4)

Expanding the Equation

Expanding the equation, we get:

y = a(x + \frac{2}{3})(x - 4)

y = a(x^2 - \frac{14}{3}x + \frac{8}{3})

y = ax^2 - \frac{14}{3}ax + \frac{8}{3}a

Comparing with the Options

Now, we can compare the expanded equation with the options given:

A. $y = 3x^2 + 10x - 8$

B. $y = 2x^2 - 5x - 12$

C. $y = 3x^2 - 10x - 8$

D. $y = 2x^2 + 5x - 12$

Option A

Comparing the expanded equation with option A, we can see that:

a = 3

-\frac{14}{3}a = 10

\frac{8}{3}a = -8

The first two equations are inconsistent, so option A is not the correct answer.

Option B

Comparing the expanded equation with option B, we can see that:

a = 2

-\frac{14}{3}a = -5

\frac{8}{3}a = -12

The first two equations are consistent, and the third equation is also consistent. Therefore, option B is a possible answer.

Option C

Comparing the expanded equation with option C, we can see that:

a = 3

-\frac{14}{3}a = -10

\frac{8}{3}a = -8

The first two equations are inconsistent, so option C is not the correct answer.

Option D

Comparing the expanded equation with option D, we can see that:

a = 2

-\frac{14}{3}a = 5

\frac{8}{3}a = -12

The first two equations are inconsistent, so option D is not the correct answer.

Conclusion

Based on the analysis, we can conclude that the quadratic equation that has solutions of $\frac{-2}{3}$ and 4 is:

y = 2x^2 - 5x - 12

This is option B.

Final Answer

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In our previous article, we discussed how to identify the quadratic equation that has solutions of $\frac{-2}{3}$ and 4. In this article, we will provide a Q&A guide to help you understand quadratic equations better.

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 is two. It can be written in the general form:

ax^2 + bx + c = 0

where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that the solutions to a quadratic equation of the form $ax^2 + bx + c = 0$ are given by:

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

Q: How do I identify the quadratic equation that has solutions of $\frac{-2}{3}$ and 4?

A: To identify the quadratic equation that has solutions of $\frac{-2}{3}$ and 4, you can use the fact that if $r$ is a solution to a quadratic equation, then $(x - r)$ is a factor of the equation. Therefore, you can write the quadratic equation as:

y = a(x - \frac{-2}{3})(x - 4)

Q: How do I expand the equation?

A: To expand the equation, you can use the distributive property:

y = a(x + \frac{2}{3})(x - 4)

y = a(x^2 - \frac{14}{3}x + \frac{8}{3})

y = ax^2 - \frac{14}{3}ax + \frac{8}{3}a

Q: How do I compare the expanded equation with the options?

A: To compare the expanded equation with the options, you can look at the coefficients of the terms. For example, if the expanded equation is:

y = ax^2 - \frac{14}{3}ax + \frac{8}{3}a

You can compare it with the options:

A. $y = 3x^2 + 10x - 8$

B. $y = 2x^2 - 5x - 12$

C. $y = 3x^2 - 10x - 8$

D. $y = 2x^2 + 5x - 12$

Q: What if I get inconsistent equations?

A: If you get inconsistent equations, it means that the option is not the correct answer. For example, if you compare the expanded equation with option A, you get:

a = 3

-\frac{14}{3}a = 10

\frac{8}{3}a = -8

The first two equations are inconsistent, so option A is not the correct answer.

Q: What if I get consistent equations?

A: If you get consistent equations, it means that the option is a possible answer. For example, if you compare the expanded equation with option B, you get:

a = 2

-\frac{14}{3}a = -5

\frac{8}{3}a = -12

The first two equations are consistent, and the third equation is also consistent. Therefore, option B is a possible answer.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. By understanding the quadratic formula and how to identify the quadratic equation that has solutions of $\frac{-2}{3}$ and 4, you can solve quadratic equations with ease. We hope this Q&A guide has helped you understand quadratic equations better.

Final Answer

The final answer is B.