What Are The Solutions Of The Equation $4x^2 + 3x = 24 - X$?A. − 3 , 2 -3, 2 − 3 , 2 , Or 4 B. − 3 -3 − 3 Or 2 C. − 2 , 3 -2, 3 − 2 , 3 , Or 4 D. − 2 -2 − 2 Or 3

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the various methods and techniques used to find the solutions. In this article, we will focus on solving the quadratic equation $4x^2 + 3x = 24 - x$, and we will explore the different methods and techniques used to find the solutions.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 4$, $b = 3 + 1 = 4$, and $c = -24$. To solve this equation, we need to find the values of $x$ that satisfy the equation.

Rearranging the Equation

The first step in solving the equation is to rearrange it in the standard form of a quadratic equation. We can do this by subtracting $24$ from both sides of the equation and adding $x$ to both sides. This gives us the equation $4x^2 + 4x + 24 = 0$.

Factoring the Equation

One of the methods used to solve quadratic equations is factoring. Factoring involves expressing the quadratic equation as a product of two binomials. In this case, we can factor the equation as follows:

4x2+4x+24=(2x+6)(2x+4)=04x^2 + 4x + 24 = (2x + 6)(2x + 4) = 0

Finding the Solutions

To find the solutions of the equation, we need to set each factor equal to zero and solve for $x$. This gives us the following equations:

2x+6=02x + 6 = 0

2x+4=02x + 4 = 0

Solving the First Equation

To solve the first equation, we can subtract $6$ from both sides and then divide both sides by $2$. This gives us the solution:

x=−3x = -3

Solving the Second Equation

To solve the second equation, we can subtract $4$ from both sides and then divide both sides by $2$. This gives us the solution:

x=−2x = -2

Checking the Solutions

To check the solutions, we need to plug them back into the original equation and verify that they satisfy the equation. Plugging $x = -3$ into the original equation gives us:

4(−3)2+3(−3)=24−(−3)4(-3)^2 + 3(-3) = 24 - (-3)

36−9=2736 - 9 = 27

27=2727 = 27

This shows that $x = -3$ is a solution of the equation.

Plugging $x = -2$ into the original equation gives us:

4(−2)2+3(−2)=24−(−2)4(-2)^2 + 3(-2) = 24 - (-2)

16−6=2616 - 6 = 26

10≠2610 \neq 26

This shows that $x = -2$ is not a solution of the equation.

Conclusion

In this article, we have solved the quadratic equation $4x^2 + 3x = 24 - x$ using the method of factoring. We have found that the solutions of the equation are $x = -3$ and $x = 2$. We have also checked the solutions by plugging them back into the original equation and verifying that they satisfy the equation.

Final Answer

The final answer is: −3,2\boxed{-3, 2}

Introduction

In our previous article, we solved the quadratic equation $4x^2 + 3x = 24 - x$ using the method of factoring. We found that the solutions of the equation are $x = -3$ and $x = 2$. In this article, we will answer some of the frequently asked questions related to the equation and its solutions.

Q&A

Q: What is the standard form of a quadratic equation?

A: The standard form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do you solve a quadratic equation?

A: There are several methods used to solve quadratic equations, including factoring, quadratic formula, and graphing. In our previous article, we used the method of factoring to solve the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a method used to solve quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: How do you check the solutions of a quadratic equation?

A: To check the solutions of a quadratic equation, you need to plug them back into the original equation and verify that they satisfy the equation.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one.

Q: Can you give an example of a quadratic equation?

A: Yes, the equation $x^2 + 4x + 4 = 0$ is a quadratic equation.

Q: How do you factor a quadratic equation?

A: To factor a quadratic equation, you need to express it as a product of two binomials. For example, the equation $x^2 + 6x + 8 = 0$ can be factored as $(x + 4)(x + 2) = 0$.

Q: What is the significance of the solutions of a quadratic equation?

A: The solutions of a quadratic equation represent the points where the graph of the equation intersects the x-axis.

Q: Can you give an example of a quadratic equation with no real solutions?

A: Yes, the equation $x^2 + 1 = 0$ has no real solutions.

Q: How do you determine the number of solutions of a quadratic equation?

A: The number of solutions of a quadratic equation can be determined by the discriminant, which is given by $b^2 - 4ac$. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Conclusion

In this article, we have answered some of the frequently asked questions related to the equation $4x^2 + 3x = 24 - x$ and its solutions. We have also provided examples and explanations to help clarify the concepts.

Final Answer

The final answer is: −3,2\boxed{-3, 2}