Solve The Equation: − X 2 − 1.5 X + 5 ≡ 7 -x^2 - 1.5x + 5 \equiv 7 − X 2 − 1.5 X + 5 ≡ 7

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: $-x^2 - 1.5x + 5 \equiv 7$. We will break down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its structure. The given equation is a quadratic equation in the form of $ax^2 + bx + c \equiv d$, where $a = -1$, $b = -1.5$, $c = 5$, and $d = 7$. Our goal is to find the value of $x$ that satisfies this equation.

Step 1: Rearrange the Equation

To make the equation more manageable, let's rearrange it to the standard form of a quadratic equation: $ax^2 + bx + c = 0$. We can do this by subtracting $d$ from both sides of the equation:

$$-x^2 - 1.5x + 5 - 7 \equiv 0$$

This simplifies to:

$$-x^2 - 1.5x - 2 \equiv 0$$

Step 2: Multiply Both Sides by -1

To make the coefficient of $x^2$ positive, let's multiply both sides of the equation by $-1$:

$$x^2 + 1.5x + 2 \equiv 0$$

Step 3: Factor the Quadratic Expression

Now that we have a quadratic expression in the form of $x^2 + bx + c$, let's try to factor it. Unfortunately, this expression does not factor easily, so we will need to use other methods to solve it.

Step 4: Use the Quadratic Formula

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

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

In our case, $a = 1$, $b = 1.5$, and $c = 2$. Plugging these values into the formula, we get:

$$x = \frac{-1.5 \pm \sqrt{(1.5)^2 - 4(1)(2)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-1.5 \pm \sqrt{2.25 - 8}}{2}$$

This simplifies to:

$$x = \frac{-1.5 \pm \sqrt{-5.75}}{2}$$

Step 5: Simplify the Square Root

The expression under the square root is negative, which means that the square root is an imaginary number. We can simplify this expression by using the fact that $\sqrt{-1} = i$, where $i$ is the imaginary unit.

$$x = \frac{-1.5 \pm i\sqrt{5.75}}{2}$$

Step 6: Simplify the Expression

We can simplify the expression by combining the real and imaginary parts:

$$x = \frac{-1.5}{2} \pm i\frac{\sqrt{5.75}}{2}$$

This simplifies to:

$$x = -0.75 \pm 1.38i$$

Conclusion

In this article, we solved the quadratic equation $-x^2 - 1.5x + 5 \equiv 7$ using algebraic manipulations and the quadratic formula. We arrived at the final answer, which is a complex number in the form of $x = -0.75 \pm 1.38i$. This solution demonstrates the power of mathematical techniques in solving quadratic equations.

Real-World Applications

Quadratic equations have numerous 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 strategies.

Final Thoughts

Solving quadratic equations is a fundamental skill that has numerous applications in mathematics, science, and engineering. By using algebraic manipulations and the quadratic formula, we can solve quadratic equations and arrive at the final answer. Whether you are a student or a professional, mastering quadratic equations is essential for success in your field.

Additional Resources

For further learning, we recommend the following resources:

• Textbooks: "Algebra and Trigonometry" by Michael Sullivan, "Calculus" by Michael Spivak
• Online Courses: "Algebra" by MIT OpenCourseWare, "Calculus" by Khan Academy
• Software: "Mathematica", "Maple", "SageMath"

Frequently Asked Questions

In this article, we will address some of the most common questions related to quadratic equations. Whether you are a student or a professional, these questions and answers will help you better understand the concept of quadratic equations.

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.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

• Factoring: If the quadratic expression can be factored into the product of two binomials, you can solve the equation by setting each factor equal to zero.
• Quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form of ax^2 + bx + c = 0, the solutions are given by:

x = (-b ± √(b^2 - 4ac)) / 2a

• Graphing: You can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

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 = (-b ± √(b^2 - 4ac)) / 2a

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

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. Then, simplify the expression under the square root and solve for x.

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. In other words, a quadratic equation has a squared variable (x^2), while a linear equation does not.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula provides two possible values for x, and these values are the only solutions to the equation.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This occurs when the expression under the square root in the quadratic formula is negative, which means that the equation has no real solutions.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression under the square root in the quadratic formula, which is given by:

b^2 - 4ac

The discriminant determines the nature of the solutions to the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: Can a quadratic equation be used to model real-world problems?

A: Yes, quadratic equations can be used to model a wide range of real-world problems, 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 strategies.

Conclusion

In this article, we have addressed some of the most common questions related to quadratic equations. Whether you are a student or a professional, these questions and answers will help you better understand the concept of quadratic equations and how to apply them to real-world problems.

Additional Resources

For further learning, we recommend the following resources:

• Textbooks: "Algebra and Trigonometry" by Michael Sullivan, "Calculus" by Michael Spivak
• Online Courses: "Algebra" by MIT OpenCourseWare, "Calculus" by Khan Academy
• Software: "Mathematica", "Maple", "SageMath"

By following these resources and practicing regularly, you can master quadratic equations and unlock the secrets of mathematics.