Comparison with the Given Solution

Oct 13, 2025 by pynkocean.com 140 views

Is the Solution Shown Below Correct? Explain

In mathematics, solving quadratic equations is a fundamental concept that is often taught in high school and early college mathematics courses. Quadratic equations are equations of the form ax^2 + bx + c = 0, where a, b, and c are constants, and x is the variable. Solving quadratic equations involves finding the values of x that satisfy the equation. In this article, we will examine a solution to a quadratic equation and determine whether it is correct or not.

The Quadratic Equation

The quadratic equation given in the solution is:

9x + 2 = 8x^2 + 6x

This equation can be rewritten in the standard form of a quadratic equation as:

8x^2 - 3x - 2 = 0

Step 1: Rearranging the Equation

The first step in solving the quadratic equation is to rearrange it in the standard form. This involves moving all the terms to one side of the equation:

8x^2 - 3x - 2 = 0

Step 2: Factoring the Equation

Unfortunately, this equation cannot be factored easily, so we will need to use other methods to solve it.

Step 3: Using the Quadratic Formula

The quadratic formula is a method for solving quadratic equations that cannot be factored easily. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 8, b = -3, and c = -2. Plugging these values into the quadratic formula, we get:

x = (3 ± √((-3)^2 - 4(8)(-2))) / (2(8)) x = (3 ± √(9 + 64)) / 16 x = (3 ± √73) / 16

Comparison with the Given Solution

The given solution is:

x = (-3 ± √(-55)) / -16

This solution is incorrect. The correct solution is:

x = (3 ± √73) / 16

Discussion

The given solution is incorrect because it involves taking the square root of a negative number. In mathematics, the square root of a negative number is not a real number, and therefore, it cannot be used as a solution to a quadratic equation. The correct solution, on the other hand, involves taking the square root of a positive number, which is a real number.

Conclusion

In conclusion, the solution shown below is incorrect. The correct solution to the quadratic equation 8x^2 - 3x - 2 = 0 is x = (3 ± √73) / 16. This solution involves taking the square root of a positive number, which is a real number. The given solution, on the other hand, involves taking the square root of a negative number, which is not a real number.

Common Mistakes

There are several common mistakes that can be made when solving quadratic equations. Some of these mistakes include:

Taking the square root of a negative number: This is a common mistake that can be made when solving quadratic equations. It is essential to ensure that the square root of a negative number is not used as a solution.
Not checking the solutions: It is essential to check the solutions to a quadratic equation to ensure that they are real numbers.
Not using the correct formula: The quadratic formula is a powerful tool for solving quadratic equations. It is essential to use the correct formula to ensure that the solutions are accurate.

Final Thoughts

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task. In our previous article, we examined a solution to a quadratic equation and determined whether it was correct or not. In this article, we will answer some frequently asked questions about quadratic equations and provide additional information to help you better understand this topic.

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, and x is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including factoring, using the quadratic formula, and completing the square. The method you choose will depend on the specific equation and the type of solution you are looking for.

Q: What is the quadratic formula?

A: The quadratic formula is a method for solving quadratic equations that cannot be factored easily. The quadratic formula is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

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

Q: What is the difference between the quadratic formula and the square root formula?

A: The quadratic formula and the square root formula are two different formulas that are used to solve quadratic equations. The quadratic formula is used to solve quadratic equations of the form ax^2 + bx + c = 0, while the square root formula is used to solve quadratic equations of the form x^2 + bx + c = 0.

Q: Can I use the quadratic formula to solve a quadratic equation with a negative leading coefficient?

A: Yes, you can use the quadratic formula to solve a quadratic equation with a negative leading coefficient. However, you will need to take the square root of a negative number, which will result in complex solutions.

Q: What are complex solutions?

A: Complex solutions are solutions to a quadratic equation that involve the square root of a negative number. These solutions are not real numbers, but rather complex numbers that have both real and imaginary parts.

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

A: To determine whether a quadratic equation has real or complex solutions, you can use 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: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to solve quadratic equations.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not checking the solutions: It is essential to check the solutions to a quadratic equation to ensure that they are real numbers.
Not using the correct formula: The quadratic formula is a powerful tool for solving quadratic equations. It is essential to use the correct formula to ensure that the solutions are accurate.
Taking the square root of a negative number: This is a common mistake that can be made when solving quadratic equations. It is essential to ensure that the square root of a negative number is not used as a solution.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task. By understanding the quadratic formula and the different methods for solving quadratic equations, you can better navigate this topic and become more confident in your ability to solve quadratic equations. Remember to check your solutions, use the correct formula, and avoid taking the square root of a negative number.