Which Equation Has One Real Solution?A. $x^2 + 2x + 5 = 0$ B. $2x^2 + 5x + 3 = 0$ C. $2x^2 - 2x + 1 = 0$ D. $x^2 - 2x - 1 = 0$

In this article, we will explore the concept of real solutions in quadratic equations and determine which of the given equations has one real solution.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable 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. The solutions to a quadratic equation are the values of x that satisfy the equation.

Real Solutions

A real solution is a value of x that is a real number, as opposed to a complex number. In other words, a real solution is a value of x that can be expressed as a decimal or a fraction.

Determining the Number of Real Solutions

To determine the number of real solutions of a quadratic equation, we can use the discriminant, which is the expression under the square root in the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

The discriminant is given by the expression b^2 - 4ac. 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.

Analyzing the Given Equations

Let's analyze the given equations and determine which one has one real solution.

Equation A: $x^2 + 2x + 5 = 0$

To determine the number of real solutions of this equation, we need to calculate the discriminant:

b^2 - 4ac = 2^2 - 4(1)(5) = 4 - 20 = -16

Since the discriminant is negative, this equation has no real solutions.

Equation B: $2x^2 + 5x + 3 = 0$

To determine the number of real solutions of this equation, we need to calculate the discriminant:

b^2 - 4ac = 5^2 - 4(2)(3) = 25 - 24 = 1

Since the discriminant is positive, this equation has two distinct real solutions.

Equation C: $2x^2 - 2x + 1 = 0$

To determine the number of real solutions of this equation, we need to calculate the discriminant:

b^2 - 4ac = (-2)^2 - 4(2)(1) = 4 - 8 = -4

Since the discriminant is negative, this equation has no real solutions.

Equation D: $x^2 - 2x - 1 = 0$

To determine the number of real solutions of this equation, we need to calculate the discriminant:

b^2 - 4ac = (-2)^2 - 4(1)(-1) = 4 + 4 = 8

Since the discriminant is positive, this equation has two distinct real solutions.

Conclusion

Based on the analysis of the given equations, we can conclude that none of the equations have one real solution. However, we can modify Equation D to have one real solution by changing the constant term.

Modified Equation D: $x^2 - 2x - 2 = 0$

To determine the number of real solutions of this equation, we need to calculate the discriminant:

b^2 - 4ac = (-2)^2 - 4(1)(-2) = 4 + 8 = 12

Since the discriminant is positive, this equation has two distinct real solutions. However, we can modify the equation to have one real solution by changing the constant term.

Modified Equation D: $x^2 - 2x + 1 = 0$

To determine the number of real solutions of this equation, we need to calculate the discriminant:

b^2 - 4ac = (-2)^2 - 4(1)(1) = 4 - 4 = 0

Since the discriminant is zero, this equation has one real solution.

Conclusion

In conclusion, the modified Equation D: $x^2 - 2x + 1 = 0$ has one real solution. This equation can be solved using the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

x = (2 ± √(0)) / 2(1)

x = 1

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In this article, we will answer some of the most frequently asked questions related to quadratic equations and real solutions.

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 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: What is a real solution?

A: A real solution is a value of x that is a real number, as opposed to a complex number. In other words, a real solution is a value of x that can be expressed as a decimal or a fraction.

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

A: To determine the number of real solutions of a quadratic equation, you can use the discriminant, which is the expression under the square root in the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

The discriminant is given by the expression b^2 - 4ac. 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: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula:

b^2 - 4ac

It is used to determine the number of real solutions of a quadratic equation.

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you need to substitute the values of a, b, and c into the expression b^2 - 4ac.

Q: What happens if the discriminant is negative?

A: If the discriminant is negative, the equation has no real solutions. This means that the equation has complex solutions, which are not real numbers.

Q: What happens if the discriminant is zero?

A: If the discriminant is zero, the equation has one real solution. This means that the equation has a single solution, which is a real number.

Q: What happens if the discriminant is positive?

A: If the discriminant is positive, the equation has two distinct real solutions. This means that the equation has two separate solutions, which are real numbers.

Q: How do I solve a quadratic equation with one real solution?

A: To solve a quadratic equation with one real solution, you can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Since the discriminant is zero, the equation has only one solution, which is:

x = (-b) / 2a

Q: Can I have multiple real solutions?

A: Yes, it is possible to have multiple real solutions. This occurs when the discriminant is positive, and the equation has two distinct real solutions.

Q: Can I have no real solutions?

A: Yes, it is possible to have no real solutions. This occurs when the discriminant is negative, and the equation has complex solutions.

Conclusion

In conclusion, we have answered some of the most frequently asked questions related to quadratic equations and real solutions. We hope that this article has provided you with a better understanding of these concepts and how to apply them to solve quadratic equations.