Solve For \[$x\$\]:$\[ X^2 - 3x = 28 \\]

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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 a specific quadratic equation, $x^2 - 3x = 28$, and provide a step-by-step guide on how to find the value of $x$. We will also discuss the importance of quadratic equations and their applications in various fields.

What are Quadratic Equations?

Quadratic equations are a type of polynomial equation that contains a squared variable, usually $x$, and can be written in the general form $ax^2 + bx + c = 0$. The equation $x^2 - 3x = 28$ is a quadratic equation because it contains a squared variable, $x$, and can be rewritten in the general form $x^2 - 3x - 28 = 0$.

The Importance of Quadratic Equations

Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of electrical circuits. Quadratic equations are also used in cryptography, coding theory, and machine learning.

Solving Quadratic Equations

There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In this article, we will focus on using the quadratic formula to solve the equation $x^2 - 3x = 28$.

The Quadratic Formula

The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Applying the Quadratic Formula

To apply the quadratic formula, we need to identify the coefficients $a$, $b$, and $c$ in the equation $x^2 - 3x = 28$. We can rewrite the equation as $x^2 - 3x - 28 = 0$, which gives us $a = 1$, $b = -3$, and $c = -28$.

Now, we can plug these values into the quadratic formula:

$$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-28)}}{2(1)}$$

Simplifying the expression, we get:

$$x = \frac{3 \pm \sqrt{9 + 112}}{2}$$

$$x = \frac{3 \pm \sqrt{121}}{2}$$

$$x = \frac{3 \pm 11}{2}$$

This gives us two possible solutions for $x$:

$$x = \frac{3 + 11}{2} = 7$$

$$x = \frac{3 - 11}{2} = -4$$

Conclusion

In this article, we have solved the quadratic equation $x^2 - 3x = 28$ using the quadratic formula. We have also discussed the importance of quadratic equations and their applications in various fields. Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike.

Additional Resources

For more information on quadratic equations and their applications, please refer to the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Frequently Asked Questions

• Q: What is a quadratic equation? A: A quadratic equation is a type of polynomial equation that contains a squared variable, usually $x$.
• Q: How do I solve a quadratic equation? A: There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula.
• Q: What are the applications of quadratic equations? A: Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science.

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have provided a step-by-step guide on how to solve the quadratic equation $x^2 - 3x = 28$ using the quadratic formula. We hope that this article has been helpful in understanding the importance of quadratic equations and their applications in various fields.

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 provide a comprehensive Q&A section on quadratic equations, covering frequently asked questions and answers.

Q&A Section

Q: What is a quadratic equation?

A: A quadratic equation is a type of polynomial equation that contains a squared variable, usually $x$. It can be written in the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are the coefficients of the equation.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation and is given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to identify the coefficients $a$, $b$, and $c$ in the quadratic equation. You can then plug these values into the quadratic formula to find the solutions.

Q: What are the applications of quadratic equations?

A: Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring if it can be factored into the product of two binomials. For example, the equation $x^2 + 5x + 6 = 0$ can be factored as $(x + 3)(x + 2) = 0$.

Q: Can I solve a quadratic equation by completing the square?

A: Yes, you can solve a quadratic equation by completing the square if the equation can be rewritten in the form $x^2 + bx + c = 0$. Completing the square involves adding and subtracting a constant term to create a perfect square trinomial.

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

A: A quadratic equation is a type of polynomial equation that contains a squared variable, while a linear equation is a type of polynomial equation that contains only a first-degree variable. For example, the equation $x^2 + 3x + 2 = 0$ is a quadratic equation, while the equation $2x + 3 = 0$ is a linear equation.

Q: Can I solve a quadratic equation with complex solutions?

A: Yes, you can solve a quadratic equation with complex solutions using the quadratic formula. The quadratic formula will provide two complex solutions, which can be written in the form $x = a + bi$ and $x = a - bi$.

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

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

Conclusion

In this article, we have provided a comprehensive Q&A section on quadratic equations, covering frequently asked questions and answers. We hope that this article has been helpful in understanding the concepts and applications of quadratic equations.

Additional Resources

For more information on quadratic equations and their applications, please refer to the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. We hope that this article has been helpful in understanding the concepts and applications of quadratic equations.