What Are The Solutions To The Quadratic Equation $50 - X^2 = 0$?A. $x = \pm 2 \sqrt{5}$ B. $x = \pm 6 \sqrt{3}$ C. $x = \pm 5 \sqrt{2}$ D. No Real Solution

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and economics. 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 solving the quadratic equation $50 - x^2 = 0$.

Understanding the Quadratic Equation

The given quadratic equation is $50 - x^2 = 0$. To solve this equation, we need to isolate the variable $x$. The first step is to add $x^2$ to both sides of the equation, which gives us $x^2 = 50$. This equation represents a parabola that opens upwards, and its vertex is at the point $(0, 50)$.

Solving the Quadratic Equation

To solve the quadratic equation $x^2 = 50$, we need to take the square root of both sides. This gives us $x = \pm \sqrt{50}$. We can simplify the square root of $50$ by expressing it as a product of its prime factors, which is $50 = 2 \cdot 5^2$. Therefore, we can write $\sqrt{50} = \sqrt{2 \cdot 5^2} = 5\sqrt{2}$.

Evaluating the Solutions

Now that we have found the solutions to the quadratic equation, we need to evaluate them. The solutions are $x = \pm 5\sqrt{2}$. We can verify these solutions by substituting them back into the original equation. If we substitute $x = 5\sqrt{2}$ into the equation $50 - x^2 = 0$, we get $50 - (5\sqrt{2})^2 = 50 - 50 = 0$. Similarly, if we substitute $x = -5\sqrt{2}$ into the equation, we also get $50 - (-5\sqrt{2})^2 = 50 - 50 = 0$. Therefore, the solutions $x = \pm 5\sqrt{2}$ are correct.

Conclusion

In conclusion, the solutions to the quadratic equation $50 - x^2 = 0$ are $x = \pm 5\sqrt{2}$. These solutions can be verified by substituting them back into the original equation. The quadratic equation $50 - x^2 = 0$ represents a parabola that opens upwards, and its vertex is at the point $(0, 50)$. The solutions $x = \pm 5\sqrt{2}$ are the points where the parabola intersects the x-axis.

Frequently Asked Questions

• What is the general form of a quadratic equation? 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.
• How do you solve a quadratic equation? To solve a quadratic equation, you need to isolate the variable $x$. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
• What is the difference between a quadratic equation and a linear equation? A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one.

Final Answer

The final answer is $x = \pm 5\sqrt{2}$.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and economics. In our previous article, we discussed the solutions to the quadratic equation $50 - x^2 = 0$. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations and their solutions.

Q&A Guide

Q1: What is a quadratic equation?

A1: 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.

Q2: How do you solve a quadratic equation?

A2: To solve a quadratic equation, you need to isolate the variable $x$. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value. You can also use the quadratic formula, which is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q3: What is the quadratic formula?

A3: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q4: How do you determine the number of solutions to a quadratic equation?

A4: The number of solutions to a quadratic equation depends on the discriminant, which is the expression $b^2 - 4ac$. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q5: What is the difference between a quadratic equation and a linear equation?

A5: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A linear equation has only one solution, while a quadratic equation can have one, two, or no real solutions.

Q6: How do you graph a quadratic equation?

A6: To graph a quadratic equation, you need to find the vertex of the parabola, which is the point where the parabola changes direction. You can find the vertex by using the formula $x = -\frac{b}{2a}$. Once you have the vertex, you can use it to graph the parabola.

Q7: What is the significance of the discriminant in a quadratic equation?

A7: The discriminant is the expression $b^2 - 4ac$ in a quadratic equation. It determines the number of solutions to the equation. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q8: Can a quadratic equation have complex solutions?

A8: Yes, a quadratic equation can have complex solutions. If the discriminant is negative, the equation has no real solutions, but it can have complex solutions.

Q9: How do you solve a quadratic equation with complex solutions?

A9: To solve a quadratic equation with complex solutions, you need to use the quadratic formula and take the square root of the negative discriminant. This will give you two complex solutions.

Q10: What is the final answer to the quadratic equation $50 - x^2 = 0$?

A10: The final answer to the quadratic equation $50 - x^2 = 0$ is $x = \pm 5\sqrt{2}$.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we provided a comprehensive Q&A guide to help you understand quadratic equations and their solutions. We hope this guide has been helpful in answering your questions and providing a deeper understanding of quadratic equations.

Frequently Asked Questions

• What is the general form of a quadratic equation? 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.
• How do you solve a quadratic equation? To solve a quadratic equation, you need to isolate the variable $x$. This can be done by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.
• What is the difference between a quadratic equation and a linear equation? A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one.

Final Answer

The final answer is $x = \pm 5\sqrt{2}$.