Solve The Equation For { X $} : : : { 5 - 7x^2 = -30 \}

Mar 9, 2025 by ADMIN 56 views

Solving Quadratic Equations: A Step-by-Step Guide to Finding the Value of x

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, 5 - 7x^2 = -30, to find the value of x. We will break down the solution into manageable steps, using algebraic techniques and mathematical concepts to arrive at the final answer.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. In our given equation, 5 - 7x^2 = -30, we can rewrite it in the standard form as 7x^2 + 30 = 5.

Rearranging the Equation

To solve the equation, we need to isolate the variable x. The first step is to rearrange the equation to get all the terms on one side. We can do this by subtracting 5 from both sides of the equation:

7x^2 + 30 - 5 = 5 - 5

This simplifies to:

7x^2 + 25 = 0

Isolating the Variable

Now that we have the equation in the standard form, we can isolate the variable x. To do this, we need to get rid of the constant term on the left-hand side. We can do this by subtracting 25 from both sides of the equation:

7x^2 = -25

Dividing Both Sides

Next, we need to get rid of the coefficient of x^2, which is 7. We can do this by dividing both sides of the equation by 7:

x^2 = -25/7

Taking the Square Root

Now that we have the equation in the form x^2 = -25/7, we can take the square root of both sides to find the value of x. However, we need to be careful when taking the square root of a negative number, as it will result in a complex number.

x = ±√(-25/7)

Simplifying the Square Root

To simplify the square root, we can rewrite it as:

x = ±√(-25)/√7

Rationalizing the Denominator

To rationalize the denominator, we can multiply both the numerator and denominator by √7:

x = ±√(-25)√7/√7√7

This simplifies to:

x = ±√(-25)√7/7

Simplifying the Square Root of -25

The square root of -25 can be simplified as:

√(-25) = √(-1)√25

Since √(-1) is equal to i (the imaginary unit), we can rewrite the equation as:

x = ±i√25/7

Simplifying the Square Root of 25

The square root of 25 is equal to 5, so we can rewrite the equation as:

x = ±i5/7

Simplifying the Complex Number

To simplify the complex number, we can rewrite it as:

x = ±(5i/7)

Conclusion

In this article, we solved the quadratic equation 5 - 7x^2 = -30 to find the value of x. We broke down the solution into manageable steps, using algebraic techniques and mathematical concepts to arrive at the final answer. The solution involved rearranging the equation, isolating the variable, dividing both sides, taking the square root, simplifying the square root, rationalizing the denominator, and simplifying the complex number. The final answer is x = ±(5i/7).

Frequently Asked Questions

What is a quadratic equation? A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, x) is two.
How do I solve a quadratic equation? To solve a quadratic equation, you need to isolate the variable x by rearranging the equation, dividing both sides, taking the square root, and simplifying the square root.
What is the difference between a real number and a complex number? A real number is a number that can be expressed as a decimal or fraction, while a complex number is a number that has both real and imaginary parts.

Additional Resources

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

Final Thoughts

Solving quadratic equations is an essential skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations and find the value of x. Remember to always check your work and simplify your answers to ensure accuracy. With practice and patience, you can become proficient in solving quadratic equations and tackle more complex mathematical problems.

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task. In our previous article, we solved the quadratic equation 5 - 7x^2 = -30 to find the value of x. However, we understand that there may be many questions and doubts that readers may have. In this article, we will address some of the frequently asked questions and provide answers to help clarify any confusion.

Q&A: Quadratic Equations

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 (in this case, 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: To solve a quadratic equation, you need to isolate the variable x by rearranging the equation, dividing both sides, taking the square root, and simplifying the square root. You can use algebraic techniques and mathematical concepts to arrive at the final answer.

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed as a decimal or fraction, while a complex number is a number that has both real and imaginary parts. In the case of quadratic equations, complex numbers can arise when the equation has no real solutions.

Q: How do I determine if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to examine the discriminant (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 two complex solutions.

Q: What is the formula for the discriminant?

A: The formula for the discriminant is b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.

Q: How do I use the discriminant to determine the nature of the solutions?

A: To use the discriminant to determine the nature of the solutions, you need to examine the value of the discriminant. 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 two complex solutions.

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: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable x by adding or subtracting the same value from both sides of the equation. You can use algebraic techniques and mathematical concepts to arrive at the final answer.

Q: What is the formula for the solution of a linear equation?

A: The formula for the solution of a linear equation is x = -b/a, where a and b are the coefficients of the linear equation.

Conclusion

In this article, we addressed some of the frequently asked questions and provided answers to help clarify any confusion. We hope that this Q&A article has been helpful in understanding quadratic equations and how to solve them. Remember to always check your work and simplify your answers to ensure accuracy. With practice and patience, you can become proficient in solving quadratic equations and tackle more complex mathematical problems.

Additional Resources

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