Consider The Equation $7x + 2 = -x^2$.What Is The Discriminant? Enter Your Answer In The Box. $\square$What Statement Is True? A. There Is 1 Rational Solution. B. There Are 2 Rational Solutions. C. There Are 2 Irrational Solutions.

Feb 26, 2025 by ADMIN 236 views

**Solving Quadratic Equations: Understanding the Discriminant**

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and physics. 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. In this article, we will focus on the equation $7x + 2 = -x^2$ and explore the concept of the discriminant.

The Discriminant

The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the symbol $\Delta$ or $D$ . The discriminant is used to determine the nature of the solutions of 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.

Calculating the Discriminant

To calculate the discriminant of the equation $7x + 2 = -x^2$ , we need to rewrite the equation in the standard form $ax^2 + bx + c = 0$ . We can do this by subtracting $7x$ from both sides and adding $x^2$ to both sides. This gives us the equation $x^2 + 7x + 2 = 0$ . Now, we can calculate the discriminant using the formula $\Delta = b^2 - 4ac$ .

import math

# Define the coefficients of the quadratic equation
a = 1
b = 7
c = 2

# Calculate the discriminant
discriminant = b**2 - 4*a*c

print("The discriminant is:", discriminant)

Interpreting the Discriminant

Now that we have calculated the discriminant, we need to interpret the result. 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.

Conclusion

In conclusion, the discriminant is a valuable tool for determining the nature of the solutions of a quadratic equation. By calculating the discriminant, we can determine whether the equation has two distinct real solutions, one real solution, or no real solutions. In this article, we have explored the concept of the discriminant and calculated the discriminant of the equation $7x + 2 = -x^2$ . We have also interpreted the result and concluded that the equation has no real solutions.

What Statement is True?

Based on the calculation of the discriminant, we can conclude that the statement that is true is:

C. There are 2 irrational solutions.

This is because the discriminant is negative, which means that the equation has no real solutions. However, the solutions can be complex numbers, which are irrational numbers.

Final Answer

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and physics. 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. In this article, we will provide a comprehensive guide to quadratic equations, including the concept of the discriminant, solving quadratic equations, and more.

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 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 the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the symbol $\Delta$ or $D$ . The discriminant is used to determine the nature of the solutions of the quadratic equation.

Q: How do I calculate the discriminant of a quadratic equation?

A: To calculate the discriminant of a quadratic equation, you can use the formula $\Delta = b^2 - 4ac$ . This formula can be used to determine the nature of the solutions of the quadratic equation.

Q: What does the discriminant tell me about the solutions of a quadratic equation?

A: The discriminant tells you about the nature of the solutions of a 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: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . This formula can be used to find the solutions of a quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to find the solutions of a quadratic equation. The quadratic formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of $a$ , $b$ , and $c$ into the formula. You also need to simplify the expression and solve for $x$ .

Q: What are the steps to solve a quadratic equation?

A: The steps to solve a quadratic equation are:

Write the quadratic equation in the standard form $ax^2 + bx + c = 0$ .
Calculate the discriminant using the formula $\Delta = b^2 - 4ac$ .
Use the quadratic formula to find the solutions of the quadratic equation.
Simplify the expression and solve for $x$ .

Q: What are the different types of solutions of a quadratic equation?

A: The different types of solutions of a quadratic equation are:

Two distinct real solutions
One real solution
No real solutions

Q: How do I determine the type of solution of a quadratic equation?

A: To determine the type of solution of a quadratic equation, you need to calculate the discriminant using the formula $\Delta = b^2 - 4ac$ . 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.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and physics. 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. In this article, we have provided a comprehensive guide to quadratic equations, including the concept of the discriminant, solving quadratic equations, and more. We have also answered some frequently asked questions about quadratic equations.

Final Answer

The final answer is: $\boxed{0}$